co2 storage in deep saline aquifers - uu.diva-portal.org
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ACTAUNIVERSITATIS
UPSALIENSISUPPSALA
2016
Digital Comprehensive Summaries of Uppsala Dissertationsfrom the Faculty of Science and Technology 1390
CO2 storage in deep saline aquifers
Models for geological heterogeneity and largedomains
LIANG TIAN
ISSN 1651-6214ISBN 978-91-554-9625-8urn:nbn:se:uu:diva-279382
Dissertation presented at Uppsala University to be publicly examined in Hamberg, Villavägen16, Uppsala, Friday, 16 September 2016 at 13:15 for the degree of Doctor of Philosophy. Theexamination will be conducted in English. Faculty examiner: Simon Mathias (University ofDurham).
AbstractTian, L. 2016. CO2 storage in deep saline aquifers. Models for geological heterogeneityand large domains. Digital Comprehensive Summaries of Uppsala Dissertations from theFaculty of Science and Technology 1390. 70 pp. Uppsala: Acta Universitatis Upsaliensis.ISBN 978-91-554-9625-8.
This work presents model development and model analyses of CO2 storage in deep salineaquifers. The goal has been two-fold, firstly to develop models and address the systembehaviour under geological heterogeneity, second to tackle the issues related to problem scaleas modelling of the CO2 storage systems can become prohibitively complex when large systemsare considered.
The work starts from a Monte Carlo analysis of heterogeneous 2D domains with a focus onthe sensitivity of two CO2 storage performance measurements, namely, the injectivity index(Iinj) and storage efficiency coefficient (E), on parameters characterizing heterogeneity. It isfound that E and Iinj are determined by two different parameter groups which both includecorrelation length (λ) and standard deviation (σ) of the permeability. Next, the issue of upscalingis addressed by modelling a heterogeneous system with multi-modal heterogeneity and anupscaling scheme of the constitutive relationships is proposed to enable the numerical simulationto be done using a coarser geological mesh built for a larger domain. Finally, in order tobetter address stochastically heterogeneous systems, a new method for model simulationsand uncertainty analysis based on a Gaussian processes emulator is introduced. Instead ofconventional point estimates this Bayesian approach can efficiently approximate cumulativedistribution functions for the selected outputs which are CO2 breakthrough time and its totalmass. After focusing on reservoir behaviour in small domains and modelling the heterogeneityeffects in them, the work moves to predictive modelling of large scale CO2 storage systems. Tomaximize the confidence in the model predictions, a set of different modelling approaches ofvarying complexity is employed, including a semi-analytical model, a sharp-interface verticalequilibrium (VE) model and a TOUGH2MP / ECO2N model. Based on this approach, theCO2 storage potential of two large scale sites is modelled, namely the South Scania site, Swedenand the Dalders Monocline in the Baltic Sea basin.
The methodologies developed and demonstrated in this work enable improved analyses ofCO2 geological storage at both small and large scales, including better approaches to addressmedium heterogeneity. Finally, recommendations for future work are also discussed.
Keywords: CO2, Carbon Capture Storage, Storage Capacity, Injectivity, Monte Carlo,Gaussian, Permeability, Upscaling
Liang Tian, Department of Earth Sciences, LUVAL, Villav. 16, Uppsala University, SE-75236Uppsala, Sweden.
© Liang Tian 2016
ISSN 1651-6214ISBN 978-91-554-9625-8urn:nbn:se:uu:diva-279382 (http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-279382)
To Joachim, Xiuchang and Caihong
List of papers
This thesis is based on the following papers, which are referred to in the text
by their Roman numerals.
I Tian L , Yang Z, Fagerlund F, Niemi A. Effects of permeability
heterogeneity on CO2 injectivity and storage efficiency coefficient.
Greenhouse Gases: Science and Technology. 2015. DOI:
10.1002/ghg.1540
II Yang Z, Tian L , Niemi A, Fagerlund F. Upscaling of the constitutive
relationships for CO2 migration in multimodal heterogeneous
formations. Journal of Greenhouse Gas Control. 2013;19(0):743-55.
III Tian L , Richard Wilkinson, Zhibing Yang, Fritjof Fagerlund, Henry
Power and Auli Niemi. Use of Gaussian Process Emulators for
Quantifying Uncertainty in CO2 Spreading Predictions in
Heterogeneous Media. 2016. Manuscript to be submitted to Computers& Geosciences
IV Tian L , Yang Z, Jung B, Joodaki S, Erlström M, Zhou Q and Niemi A
Integrated simulations of CO2 spreading and pressure response in the
multilayer saline aquifer of South Scania Site, Sweden. GreenhouseGases: Science and Technology. 2016. DOI: 10.1002/ghg.1583
V Yang Z, Tian L , Jung B, Joodaki S, Fagerlund F, Pasquali R, et al.
Assessing CO2 storage capacity in the Dalders Monocline of the Baltic
Sea Basin using dynamic models of varying complexity. InternationalJournal of Greenhouse Gas Control. 2015;43:149-60.
Reprints were made with permission from the publishers.
In Paper I, I designed the study with the help from other co-authors. I
performed all the numerical analyses and wrote the main part of the manuscript.
In Paper II, I carried out the numerical simulations using TOUGH2 / ECO2N
including sensitivity analyses and adapting the upscaling methodology on the
concluding benchmark problem and participated to the writing. For PaperIII, the joint effort of the original paper was initiated by Professor AndrewCliffe at the School of Mathematical Sciences, University of Nottingham, who
unfortunately passed away during the preparation of the collaborative work.
I thereafter took over the main responsibility of the work and then finished
the manuscript in collaboration with other co-workers from University of Not-
tingham, University of Sheffield and Uppsala University. I was responsible in
conducting a series of Monte Carlo simulations, using the emulator methodol-
ogy to estimate cumulative distribution functions for selected modelling out-
puts and exploring the issues encountered in adapting the methodology. In
Paper IV, I compiled all the data, built the conceptual model and wrote the
manuscript. The injectivity analysis using a semi-analytical model was done
by ZY. And the analysis using vertical equilibrium model was done by BJ.
For Paper V, I built the conceptual model, constructed a three-dimensional
model using the original digital elevation data and then performed detailed
numerical simulations for CO2 plume evolution and pressure response using a
massive parallel version of TOUGH2 as well as participated in the writing of
the manuscript.
In addition, I have contributed to the following journal publications which
are related to but not included in the thesis:
Yang Z, Niemi A, Tian L , Erlström M. Modelling of Far-field Pressure
Plumes for Carbon Dioxide Sequestration. Energy Procedia. 2013;40:472-80.
Yang Z, Niemi A, Tian L , Joodaki S, Erlström M. Modeling of pressure
build-up and estimation of maximum injection rate for geological CO2 storage
at the South Scania site, Sweden. Greenhouse Gases: Science and Technol-ogy. 2015;5(3):277-90.
Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.1 Typical Storage Scenario . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2 CO2 Migration in a Heterogeneous Reservoir . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.2.1 Types of heterogeneity to be addressed . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.2.2 Stochastic representation of a heterogeneous system . . . 22
2.2.3 Constitutive relationships in heterogeneous system . . . . . . 24
2.3 Key Issues in Predictive Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.3.1 Capacity estimates for GCS project . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.3.2 CO2 migration and injectivity analyses . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.4 Computational Challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.4.1 Heterogeneous system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.4.2 Modelling of large-scale systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3 Modelling the Effects of Geological Heterogeneity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.1 Effects of Heterogeneity on Storage Performance (Paper I) . . . . . . . 28
3.1.1 The injectivity index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.1.2 The storage efficiency coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.1.3 Flow regime analyses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.2 Upscaling of Heterogeneity (Paper II) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.2.1 Multimodal distribution of permeability . . . . . . . . . . . . . . . . . . . . . . . . 34
3.2.2 Upscaling of constitutive relationships . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.2.3 Upscaled simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.3 Gaussian Emulator Methods for Modelling a Heterogeneity
System (Paper III) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.3.1 Principle of Gaussian process emulation . . . . . . . . . . . . . . . . . . . . . . . 39
3.3.2 Parameterization and experiment design . . . . . . . . . . . . . . . . . . . . . . . 40
3.3.3 Applying Gaussian process emulation for uncertainty
analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.3.4 Comparison to the conventional Monte-Carlo
approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4 Modelling of Large CO2 Storage Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.1 Large Scale Models of Varying Complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.2 The Migration-limited System — South Scania Site (Paper IV) 47
4.2.1 Injectivity analysis for South Scania . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.2.2 Migration analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.3 The Injectivity-limited System — Dalders Monocline (Paper
V) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.3.1 Injectivity analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.3.2 CO2 plume migration and capacity limiting criteria . . . . . 53
5 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
6 Acknowledgement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
7 Sammanfattning på svenska . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
Abbreviations
CCSCarbon Capture and Storage
CRPSContinuous Rank Probability Score
CDFCumulative Distribution Function
ECDFEmpirical Cumulative Distribution Function
ECO2NA TOUGH2 fluid property module for mixtures of water, NaCl, and CO2
GCSGeological Carbon Storage
GPGaussian Processes
K-LKarhunen-Loève (decomposition)
log-GRFlogarithmic-distributed Gaussian Random Field
MCMonte Carlo (simulations)
RFRandom Function
RVRandom Variable
TOUGH2A general-purpose numerical simulation program for multi-dimensional
fluid and heat flows of multiphase, multicomponent fluid mixtures in
porous and fractured media
T2MPTOUGH2-MP-ECO2N (simulation program)
TOUGH2-MP is a massive parallel version of the TOUGH2 family of
codes
UAUncertainty Analysis
VEVertical Equilibrium (model)
Symbols
ECO2 storage efficiency coefficient [-]
FopOverpressure ratio [-]
Iin jCO2 injectivity index [kg ·Pa−1]
k(Absolute) permeability [m2]
ZRandom variable
PPressure [Pa]
Pc is capillary pressure; Pcd is a characteristic capillary pressure;
Pe is capillary entry pressure
QMass flow rate [Kg ·S−1]
SSaturation [-]
VDPDykstra-Parsons coefficient [-]
γParameter related to local-scale pore size distribution [-]
λDimensionless correlation length of the permeability fields [-];
λi stands for ordered eigenvalues in K-L decomposition
ξFitting parameter for injectivity analysis [-];
ξi stands for random variable in K-L decomposition
σStandard deviation of the isotropic absolute permeability [m2];
φLocal porosity [-];
φi stands for eigenfunctions in K-L decomposition
ΣCovariance matrix for Z specification
subscript
b breakthrough (capillary pressure)
e effective (value), equivalent to e f f
g gaseous
l liquid / aqueous
L large scale average
w wetting (phase)
r relative (value)
x,y,z x, y, z–directional
eq far-end reference (location)
ex externally applied
gr residual gas
nw non-wetting (phase)
in j injection (location)
re f reference (value)
w−b well to boundary
superscriptd dimension (integer)M,N size of data set (integer)
1. Introduction
Carbon Capture and Storage (CCS) is the processes of capturing car-
bon dioxide (CO2) emitted from large point sources, such as fossil fuel based
power plants and industrial sources, and depositing it in deep geological for-
mations. CCS provides a possibility to cut emissions while maintaining access
to fossil fuel energy until sufficient alternative energy sources exist and it is
necessary for some process industries. It is considered by e.g. the Interna-
tional Panel on Climate Change (IPCC, 2014) as an important option in the
portfolio of mitigation actions for stabilization of atmospheric greenhouse gas
concentrations. When the target storage medium is an underground geological
reservoir, such as deep saline formation, the term used for the storage part of
CCS is Geological Carbon Storage (GCS).
The appraisal of a GCS project requires trustworthy assessments such as
where the injection site shall be located and the capacity of the storage oper-
ation. This in turn requires careful site characterization as well as predictive
modelling of site performance. For Europe, this is regulated in EU directive
on the geological storage of carbon dioxide (European Parliament and Council
Directives, 2009).
Among all the assessment methodologies, numerical simulations is an
important component owing to its ability to address the relevant physical and
chemical processes at the relevant spatial and temporal scales. The under-
ground migration of CO2 is characterized by a two-phase flow system where
the injected CO2 is displacing the resident formation water. Reservoir sim-
ulation has been used to answer a wide range of questions related to GCS
and novel approaches have been developed to address specific questions to
improve the analysis.
Reliable modelling obviously critically depends on the amount of the data
available. Some of the key issues for modelling a specific site as a candidate
for GCS are:
1. the role of geological heterogeneity on model prediction and the re-
lated uncertainty;
2. how should the uncertainty be quantified;
3. challenges posed by the large size of the domains to be modelled;
at scales of kilometres (tens or even hundreds of kilometres when
pressure evolution is considered) when some of the key processes
may take place at a scale of the pores.
To address these issues, reliable approaches are needed both for dealing with
the heterogeneity as well as dealing with the large scales of the model domains.
15
Aims of the thesis
The work presented here has the objective to develop tools and method-
ologies for the characterization and modelling of deep saline aquifers for long
term storage of CO2. The objective can further be expressed by the following
research questions:
• How can the effect of geological heterogeneity be modelled and quan-
tified,and what is the effect of heterogeneity on storage performance?
• How can the prediction performance and reliability be improved, mak-
ing the best use of data,computational resources available, and with
particular focus on geological heterogeneity and the large size of the
domains that need to be modelled.
The thesis therefore focuses on the effect of formation heterogeneity and
the effective simulation methodologies resolving the underground CO2 mi-
grations at the relevant spatial and temporal scales. The specific questions
addressed are:
• How does geological heterogeneity influence CO2 underground mi-
gration and CO2 storage performance? (Paper I)
• Can upscaling of the constitutive relationships help in addressing CO2
migration in heterogeneous formations at larger scale? (Paper II)
• Can heterogeneity and the related model uncertainty be presented
with simplified modelling approaches such as by using an emulator,
that provides an approximation to the full simulator but at only a frac-
tion of the computational cost? (Paper III)
• What is the most appropriate modelling procedure for an industrial
storage scenario at a scale of tens of kilometres? (Papers IV and V)
The remaining of this text has been organized by first presenting the back-
ground to the issues encountered in the numerical simulations of geologi-
cal CO2 storage, followed by the modelling methodologies developed in this
work, along with the results and conclusions.
16
2. Background
The processes that occur upon the injection of CO2 into a deep geological
formation involve movement of two fluid phases. One is the native aqueous
phase that initially fills the void space. The other is the supercritical CO2
phase that is injected through a well. As the CO2 comes into contact with
the formation water and spreads out into the formation, it will dissolve in
the aqueous phase as well as cause chemical changes that result in carbonate
mineralization (IPCC, 2005). The migration pattern of CO2 is determined by
the characteristics of the geological media as well as the state of the injected
CO2. The effect of the reservoir rock heterogeneity on the two-phase buoyant
transport is therefore a key component influencing CO2 migration and thereby
also the subsequent processes of dissolution and mineralization
2.1 Typical Storage Scenario and CO2 TrappingProcesses
A schematic setting of CO2 injection into a storage reservoir is shown in
Figure 2.1 where CO2 is injected from a vertical well on the left hand side.
Once the supercritical CO2 has entered the target aquifer, it tends to move up-
wards due to its smaller density in comparison to that of the resident brine.
The storage reservoir is overlain by a sealing layer (caprock) of lower per-
meability, which provides both a capillary as well as a permeability barrier
preventing the lighter CO2 from rising up. The accumulation of CO2 at the
upper part of the aquifer may resemble a cone, the exact shape of the plume
being dependent on e.g. the injection rate and pressure as well as the hydraulic
conductivity of the aquifer (Nordbotten et al., 2005). When injecting CO2 into
the storage reservoir overpressure in comparison to the in-situ pressure is ap-
plied. It is important that this overpressure is within the maximum sustainable
pressure (Rutqvist et al., 2007; Mathias et al., 2009) the exceeding of which
will cause damage to the sealing layer by mechanical failure.
The process of immobilizing the injected CO2 by topographical features
of the sealing rock is called structural trapping which is a very important
mechanism in particular for some dome-shaped or other none-flat sealing units.
The injected CO2 also dissolves into the formation brine when moving
as a gas / supercritical fluid away from the injection well. This is called sol-ubility trapping and is another important trapping mechanism. This process
of dissolution can increase the density of the brine and promote convective
17
(a)
(b)Figure 2.1. Schematic of the plume shape in the simple CO2 injection problem.
mixing throughout the permeable thickness of the storage aquifer (Pruess and
Spycher, 2007), a process enhancing further dissolution (Ennis-King et al.,
2005).
After the cease of injection, the CO2-rich phase will continue to move
into the aquifer. During this transport, part of the CO2 is left behind, trapped
inside the rock pores by capillary forces. In Figure 2.1b, an imaginary line
is drawn (for visualization purpose) to distinguish the mobile CO2 and its
non-mobile part. The immobilized CO2 that is left behind is called residualtrapping, an additional important trapping mechanism. The residually trapped
CO2 is also a trace of the CO2 migration pathway. The amount of residual
trapping is determined by where the CO2 plume has visited and the properties
of the rock matrix (Rasmusson et al., 2014, 2016).
Finally, the dissolved CO2 causes chemical changes in the brine compo-
sition that with time will lead to formation of carbonate minerals (Rosenbauer
et al., 2005) which is called mineral trapping. This fourth trapping process,
however, involves very long time periods (IPCC, 2005) and, while considered
a secure final trapping form, is not important at the early stages of the project
addressed in this Thesis.
Figure 2.2 exemplifies trapping contributions from various mechanisms
for an idealized schematic case according to IPCC (2005), and Figure 2.3
shows the specific case for the South Scania site where the three trapping
contributions of structural, residual and dissolution trapping as a function of
time have been estimated by means of numerical modelling (Paper IV). Such
18
Figure 2.2. Storage security depends on a combination of physical and geochemical
trapping. Over time, the physical process of residual CO2 trapping and geochemical
processes of solubility trapping and mineral trapping increase (IPCC, 2005).
Figure 2.3. CO2 trapping contribution (percentage of the total mass) as a function of
simulation time for the the South Scania site. The total 150 Mt CO2 stored at the site
consists of three components: (1) Structural trapping ( CO2 saturation (S) > residual
saturation (Sgr)); (2) Residual trapping (S≤ Sgr), if gaseous CO2 has visited a certain
pore volume; (3) Solubility trapping, i.e. the CO2 dissolved in the aqueous phase.
predictions will always be site specific and depend on the characteristics of the
site.
19
2.2 CO2 Migration in a Heterogeneous Reservoir
2.2.1 Types of heterogeneity to be addressed
Below the main types of heterogeneity that may need to be addressed
when modelling CO2 geological storage are outlined. The level of complexity
depends on the purpose and needs of the modelling study.
Layered system and layer-to-layer heterogeneityThe first level of heterogeneity is the layer-to-layer heterogeneity formed
by a succession of different sedimentary layers, where reservoir layers of dif-
ferent properties and low permeability sealing layers alternate. In modelling
such systems one may typically be, at least as the first approximation, more in-
terested in accurately describing the geometries of the layers and their average
properties, while ignoring the heterogeneity within each layer.
A good example of studies addressing such systems is that of Birkholzer
et al. (2009) which investigates pressure response caused by CO2 injection
using a very big layered domain starting from the target reservoir all the way
up to the ground surface. Chasset et al. (2011) in turn, explored the feasibility
of CO2 injection at the Scania site, Sweden using a multilayered radial (radial
2D) system. In this Thesis, this kind of heterogeneity is addressed especially
in Paper IV where a complex multi-layer 3D model of the South Scania site
is presented.
Heterogeneity described through statistical propertiesThe second type of heterogeneity is unresolved heterogeneity within a
geological unit or layer that is typically described through geostatistical mod-
elling. For such a system the properties of the medium are not known at every
point in space but are described through their geostatistical properties. The
idea is to characterize the unsampled / unknown properties such as porosity,
permeability as random variables (RVs), whose behaviour is characterized by
random function (RF). The spatial variability is inferred by kriging or esti-
mated by simulations using a variogram model (Deutsch and Journel, 1998).
Generation of random fields based on these methods provides possible real-
izations of the property distribution in space. These realizations can, in turn,
be used as input for e.g. Monte Carlo (MC) simulations. This type of hetero-
geneity is addressed in Papers I and III.
Lithofacies scale heterogeneityThe third type of heterogeneity, often encountered in cases of sedimen-
tary formations, is a heterogeneity pattern defined by the distribution of geo-
logical facies (Bloomfield et al., 2006; Lengler et al., 2010), the distribution
of which can often be defined statistically. Within them, such facies can also
contain heterogeneity of the second type discussed above. Paper II addresses
this type of heterogeneity.
20
As an example of the different type of heterogeneities we can take the South
Scania site, Sweden (Fig. 2.4) that has been studied in several papers of this
Thesis.
Figure 2.4. The map of the South Scania site (Paper IV)
The South Scania site is located in the province of Scania, Sweden (Fig.
2.4). The site has been studied for oil exploration and thermal energy produc-
tions, and an extensive geological database is available. In terms of geological
storage of CO2 the site has previously been studied by Chasset et al. (2011)
who looked at the data from borehole FFC-1 and focused on parameter sensi-
tivity effects. Extensive data analysis in terms of on CO2 injection was carried
out in the context of the EU FP7 MUSTANG project e.g. Erlström et al. (2011)
and is used as basis or background in Papers II and IV of this Thesis.
South Scania site is a heterogeneous stratigraphic system. Eight geolog-
ical units have been mapped to the 3D geological model (Fig. 2.5). The pri-
mary caprock units C1 and C2 are relatively thick (total thickness ca. 250−470m) and the strata identified have a consistent distribution. Their sealing
ability is considered to be sufficient to prevent upward CO2 migration. Below
these caprock units, the Lower Cretaceous Arnager Greensand is identified as
a secondary trap, R1. Underlying R1 are a relatively thin (ca. 30m), homo-
geneous intermediate seal C3 and the primary trap R2 which is a highly per-
meable aquifer, stratigraphically located at the Jurassic-Cretaceous transition
interval. Below R2 follows a Rhaetian-Pliensbachian multilayered sequence
of sandstone and claystone rendering two main traps, R3 and R4, separated by
21
Figure 2.5. Schematic of a vertical cross section of the 3D geological model (east-
west view). The cross section (highlighted in Fig. 2.4) passes through respectively
Kungstorp-1 (Ku-1), Eskilstorp-1 (Es-1), FFC-1 and Barseback-1 (Ba-1). The scale
in vertical direction is magnified by a factor of 25 (Paper IV)
an intermediate seal, C4. Depending on the objective of the modelling, one can
map its heterogeneous properties onto different type of models. In this The-
sis, for the GCS capacity estimate of the entire site, multiple hydro-geological
units (Fig. 2.5) are considered and modelled as internally homogeneous units
in a fully 3D model (Paper IV). A 2D multimodal model (above-mentioned
third type of heterogeneity) consisting of three lithofacies, namely sandstone,
siltstone and clay (Paper II) with their internal heterogeneity structures in
turn, is used to explore the feasibility of constitutive relationship upscaling.
The model configuration in this study (Paper II) is based on an ex-ante inter-
pretation combining units R3 and R4. The medium properties are based on
literature.
2.2.2 Stochastic representation of a heterogeneous system
The physical properties of the reservoir such as porosity and permeability
are often treated as RVs due to data scarcity (Tsang et al., 2008). They are
highly variable in space but in general not purely random (de Marsily, 1986).
That is, if the measurements are made at two different locations, the closer the
measurements are to each other the closer the measured values. This kind of
correlation is described by a variogram model (Deutsch and Journel, 1998).
A RV is traditionally denoted by a capital letter, such as X (de Marsily,
1986; Gelhar, 1993; Deutsch and Journel, 1998). In our case the property of
interest is the unresolved heterogeneous permeability field. We use Z to denote
a single random permeability field representing a possible set of permeability
values (tensors) that resemble our model domain. The lower-case k is used to
represent specific permeability tensors for each numerical grid. Z is a RF such
22
as {Z(x),x ∈model domain}, with x being the location coordinates vector. A
number of specific realizations Z1, . . . ,Zn can be acquired through e.g. generic
kriging (Deutsch and Journel, 1998) and can be conditioned on few observa-
tions otherwise controlled by simple parametric descriptions of the property
distributions (probability density function and variogram).
We model Z as a logarithm Gaussian random field (log-GRF)(Hoeksema
and Kitanidis, 1985), and assume that
logZ ∼ N(μ,Σ).
The covariance structure is specified through the matrix Σ. It is common to
specify Σ through a function, c(x,x′), that specifies the covariance between
the random field at any two locations, x and x′. In common with (Cliffe et al.,
2011), we assume an isotropic exponential covariance function
c(x,x′) = σ2 exp
(−
2
∑i=1
|xi− x′i|λi
)(2.1)
where λ= (λ1,λ2) is the characteristic length scale in each dimension (2D
for i = 1,2), and σ2 represents the variance in term of the isotropic absolute
permeability. The term |xi− x′i| is the distance between the two points in the
ith dimension.
Figure 2.6. Example log permeability fields (σ = 0.2) with different dimensionless
horizontal correlation lengths (λ ): (a) λ = 0.30; (b) λ = 0.15; (c) λ = 0.075; (d) λ =
0.02.
One realization of Z is an image showing a possible permeability dis-
tribution (Fig.2.6) that can be used as input to our CO2 injection simulation.
Such generated random permeability fields from a specified distribution are
equiprobable, meaning that any one of the realizations has the same probabil-
ity to be drawn / produced as any other from the total number of realizations
(Deutsch and Journel, 1998). If each log-GRF can be identified to a single
23
random number or vector (seed), the specific realization can be reproduced
exactly using the same algorithm. Therefore the population of the log-GRF is
algorithm dependent (depends on computer code and parameter values).
Such realizations are used to determine the probabilities of occurrence of
specific functions (simulation outputs) and can be produced in many ways
(Deutsch and Journel, 1998). In the thesis work presented here, we have
used Field Generator (Chiang, 2005) in Paper I, HydroGen (Bellin and Ru-
bin, 1996) in Paper II and a Karhunen-Loève decomposition in Paper III (see
the enclosed papers for details). Figure 2.6 presents one example showing the
use of log-GRFs to represent heterogeneity in a single rock type (facies). In
fact, multiple facies can coexist and the distribution of the facies type can also
be represented using statistical model. A multimodal heterogeneous model
including both the facies distribution and properties within the facies is dis-
cussed in Section 3.2.1 and in Paper II.
A stochastic representation of Z is considered as input to a model such
as the simulation code TOUGH2 / ECO2N denoted here by f . The numeri-
cal simulations are then used to map Z to some outputs y, for example, CO2
breakthrough time (see also Section 2.3.2), f : Z → y. The whole process can
therefore be seen as:
y = f (Z) (2.2)
2.2.3 Constitutive relationships in heterogeneous system
In two-phase flow, constitutive relationships refer to the functions used to
describe the dependency between the capillary pressure (Pc), the correspond-
ing permeability reduction, expressed by relative permeability, and phase sat-
uration (e.g. Fagerlund et al., 2006).
Traditionally, for numerical simulations of the multi-phase system, cap-
illary pressure(Pc) is related to fluid saturation (S) where the function con-
tains information about pore size distribution. The different fluids in porous
medium can obstruct the movement of each other. Such reduction is normally
expressed as a ratio of the effective permeability to the permeability in sin-
gle phase condition (relative permeability kr = ke f f /k) and is therefore also
expressed as a function of the phase saturation.
For the two-phase flow simulations using TOUGH2 / ECO2N in this the-
sis, the Van Genuchten (1980) model is used for the capillary pressure as well
as the relative permeability of the aqueous phase, and the Corey (1954) func-
tion is used for the relative permeability of the gaseous / supercritical phase.
Furthermore, an assumption is made that the local capillary entry pressure
(Pe) has the same spatial distribution as the local permeability, so that low per-
meability locations correspond to higher capillary entry pressure. For each
numerical cell the capillary entry pressure is scaled according to (Leverett,
24
1941):
Pe = Pe,re f
√kre f
k(2.3)
where Pe,re f is the reference entry pressure and the kre f is the reference per-
meability.
2.3 Key Issues in Predictive Modelling
2.3.1 Capacity estimates for GCS project
Trustworthy modelling is a key to successful GCS implementation. The
quantitative assessment of a project addresses issues such as volumes that
can be stored, the time scale that a project can last and the hydro-thermo-
mechanical-chemical responses that can be expected. An important aspect
in feasibility evaluation of a GCS project is to provide a capacity estimate,
in other words, estimate how much of CO2 can be stored and in what time
period, without any undesired effects. There are many different ways to pro-
vide capacity estimates, ranging from the simplified volumetric calculations
to the more advanced numerical simulations (Bachu and Adams, 2003; Zhou
et al., 2008; Bachu, 2015). The problem can be further complicated when het-
erogeneity is considered (Hovorka et al., 2004; Ambrose et al., 2007; Nicot,
2008; Deng et al., 2012).
For a GCS project, if the migration of CO2 becomes a major constraint
such that CO2 can reach a leakage point, the capacity is migration limited. On
the other hand, if the pressure buildup at the injection well becomes a limiting
factor and further increase of the injection rate can lead to caprock damage,
the capacity is injectivity limited. It is not possible to determine the type of
capacity for a project site beforehand (Szulczewski et al., 2012)
When estimating the storage capacity we need to identify, by means of
dynamic modelling which type of constraints will become dominant for a po-
tential GCS project.
2.3.2 CO2 migration and injectivity analyses
Storage efficiency and migration analysisThe analysis on volumetric CO2 storage efficiency (E) is based on Bachu
(2015):
E =VCO2
Vpore(2.4)
where VCO2is the volume of the injected CO2 and Vpore is the total pore volume
of the domain.
The evaluation of E depends on the time-scale, and it may vary through-
out the time it is being monitored. For example, in the analysis conducted in
25
Paper I, the breakthrough time when the front of CO2 reaches the monitoring
well is used to determine E in the model domain. More detailed characteriza-
tion of the capacity based on both plume migration and pressure evolution is
considered through 3D modelling in Paper IV and V.
Injecitivity analysisInjectivity characterizes the ease by which the fluid can be injected into a
geological formation (IPCC, 2005).
The injectivity analyses are based on the injectivity index (Iin j, based on
Law and Bachu, 1996) according to:
Iin j =Q
Pin j−Peq(2.5)
where Q is the CO2 mass flow rate at the time of breakthrough, Pin j is the
pressure at the injection borehole, Peq is the pressure at the far-end spill point.
Assuming a fixed CO2 injection rate, a high injectivity index means that less
significant pressure build-up would occur at the point of injection.
Another way to characterize injectivity is by directly using the pressure
build up caused by the CO2 injection where an overpressure factorFop is de-
fined by
Fop(%) =PF −PH
PH×100 (2.6)
where PF is the fluid pressure caused by the CO2 injection and PH is the hy-
drostatic pressure.
2.4 Computational Challenges in ModellingHeterogeneous and Large Scale Systems
The time and effort (cost) spent on producing a certain modelling output
depends on the problem setup and its objective. The computational cost of
numerical simulation is mainly dependent on the algorithm used (complexity
and solving strategy) and the size of the model domain (discretization and
resolution). Both issues are addressed in this Thesis.
2.4.1 Heterogeneous system
When addressing the sensitivity of CO2 storage performance on the het-
erogeneity characteristics of the model domain (Paper I) a large number of
heterogeneous permeability realizations need to be modelled. In Paper I this
is done with a traditional Monte Carlo methodology using TOUGH2/ECO2N.
Depending on e.g. the model size, such Monte Carlo simulations may become
too expensive as a large number of realizations is needed (the accuracy of the
26
result depends on the number of runs). To address this issue, in Paper III,
a computationally more economical approach based on a Gaussian Process
Emulator (GPE) is introduced and tested.
2.4.2 Modelling of large-scale systems
Site characterization typically involves modelling of large scale systems.
Depending on the availability of the computational resources and the accuracy
aimed at, difference strategies can be applied.
In Paper II, an upscaling method is presented so that a bigger domain can
be modelled using a relative coarse upscaled model, to get a rough estimate of
the model behaviour.
For more detailed predictive modelling of site-specific CO2 storage po-
tential, large 3D models are needed. The 3D South Scania site model for ex-
ample consists of more than 70,000 elements (Paper IV). For the modelling
of Dalders Monocline (Paper V), the entire domain is discretized into over
200,000 elements. To handle the large number of elements, parallel method-
ologies implemented in TOUGH2-MP (T2MP) are used where the simula-
tion domain is partitioned into several sub-domains so that the simulations are
run as multiple processes on a few or many processors (CPU) simultaneously
(Zhang et al., 2008).
Due to the large computational expense of the fully 3D simulations, we
also introduce a step-wise modelling approach where the use of the most de-
manding simulation is minimized, by first making preliminary estimates of
pressure response and migration with more simpler models, including a semi-
analytical model (Mathias et al., 2009; Yang et al., 2015) and a sharp interface
vertical equilibrium (VE) model (Bear, 1972; Nordbotten et al., 2005; Gasda
et al., 2009) as well as the aforementioned "full physics" TOUGH2 / ECO2N
model. The gain in modelling resolution and accuracy is presented by stepwise
adapting models of increasing complexity (Paper IV and V).
27
3. Modelling the Effects of GeologicalHeterogeneity
The task for Chapter 3 is to answer what kind of output y would be pro-
duced by a model for what kind of heterogeneity in its inputs. Our analyses are
through evaluating numerical models at selected outputs y1 = f (Z1), . . . ,yn =f (Zn) and then to use {y1, . . . ,yn} to derive specific quantities of interests.
Paper I looks at the effects of varying parametric descriptors to the het-
erogeneous permeability fields and how the change of correlation structure
will change the CO2 storage capacity estimates. Paper II looks at multimodal
heterogeneity and explores a methodology of using multiple detailed models
to derive equivalent properties at coarser scale for applications in larger scale
model. Paper III proposes an alternative approach using Gaussian process
emulator and further explores the forward propagation of uncertainty through
modelling.
3.1 Effects of Heterogeneity on Storage Performance(Paper I)
The heterogeneity explored in Paper I is the type of permeability that can
exist within a thin sandstone reservoir layer (Fig.3.1). Multiple realizations
(see Fig.2.6 for example) are generated in the software Field Generator (Chi-
ang, 2005) using a controlled variogram model. By systematically varying λand σ (Eq.2.1), 16 combinations are chosen, for each of which 100 realiza-
tions of Z are generated. For further evaluation of the permeability variation,
Dykstra-Parsons coefficient (Dykstra-Parsons coefficient) is used in a follow-
up flow regime analysis as suggested by Waggoner et al. (1992) based on:
VDP =k50− k84.1
k50(3.1)
where k50 is the median permeability and k84.1 is the permeability at one stan-
dard deviation below the median value.
The numerical simulations are performed using the TOUGH2 code (Pruess
et al., 1999), a highly established general-purpose simulator for non-isothermal
multiphase flow in porous and fractured media, with the equation-of-state
module ECO2N for CO2 and brine at reservoir conditions (Pruess and Spy-
cher, 2007).
28
Figure 3.1. Conceptual model for a 2D CO2 storage aquifer simulation
The effects of permeability heterogeneity on storage performance are an-
alyzed using the injectivity index and storage efficiency coefficient as mea-
sures (see Section 2.3.2 for definitions).
3.1.1 The injectivity index
The CO2 injection simulations were carried out for the 16 parameter com-
binations using various injection pressures, each with 100 permeability field
realizations. The injectivity index is taken at the time of the CO2 breakthrough
at the right hand boundary (Fig.3.1). For a given standard deviation the injec-
tivity index increases with the horizontal correlation length, the effect being
bigger the larger the standard deviation is (Paper I). The difference between
realizations increases as both λ and σ increase. Increasing σ results in lower
injectivity for small correlation lengths, but leads to higher injectivity for large
correlation lengths.
The detailed effects of λ and σ , as well as their interplay on injectivity
has been given in Paper I. It is of interest to find an overall model capturing the
effect, where we can see that the effect of correlation length depends on the
magnitude of standard deviation. Model testing revealed that we could find an
excellent linear fit between the mean injectivity index Iin j and the parameter
group (λ/ξ )σ as shown in Figure 3.2. In this group the additional parameter
ξ that is used to scale the correlation length is found to be ξ = 0.12, in our
particular model. This term can be expected to be case dependent and vary
according to factors such as domain and injection geometry, flow regime etc.
The possible physical explanations to this factor are further discussed below.
3.1.2 The storage efficiency coefficient
In a similar way, the storage efficiency coefficient (E) is also calculated
and plotted as a function of λ and σ and an injection pressure factor. On
average, both increasing λ and increasing σ result in smaller E. The injec-
tion pressure slightly affects E when σ is large but the effect is much less
29
Figure 3.2. Injectivity index (Iin j) as a function of parameter group (λ/ξ )σ
Figure 3.3. Storage efficiency coefficient (E) as a function of the parameter group
λσ 2
30
pronounced in comparison to the reducing effect of the heterogeneity parame-
ters (λ and σ together).
As the case of injectivity above, a general dependency between storage
efficiency coefficient and parameters describing medium heterogeneity is use-
ful. When plotting the mean storage efficiency coefficient for all the cases as
a function of the parameter group λσ2, a clear relationship with a roughly
linear fit can be obtained (Fig.3.3). The parameter group proposed here is the
same as the heterogeneity index suggested by Waggoner et al. (1992) to be
used to address dispersive displacement during the EOR operations. It is also
equal to (when divided by a constant flow domain dependent factor) what is
known as macrodispersivity in the theory of solute transport in groundwater
flow (Gelhar and Axness, 1983).
3.1.3 Flow regime analyses
To inspect the behavior of injectivity and storage efficiency coefficient as
a function of the prevailing flow regime, the mean injectivity index and the
storage efficiency coefficient from the simulations were plotted on graphs of
the type suggested by Waggoner et al. (1992) with Dykstra-Parsons coefficient
on the x-axis and dimensionless correlation length on the y-axis. The Dykstra-
Parsons coefficient is a measure of permeability variation (see Eq.3.1).
Figure 3.4. Injectivity index (Iin j) as a function of correlation length (λ ) and Dykstra-
Parsons coefficient (VDP). For flow regime analysis: the contour lines show the transi-
tion from lower Iin j region (lower right) to the higher Iin j region (upper right)
The result for injectivity index is shown in Figure 3.4. On the plot the
region in the lower right-hand-side the flow is what can be called disper-
sive (large σ , small λ ) and moving up and towards the upper left-hand-side
31
Figure 3.5. Storage efficiency coefficient (E) as a function of correlation length (λ )
and Dykstra-Parsons coefficient (VDP). For flow regime analysis: the contour lines
show the transition from lower E region (upper right) to the higher E region (lower
left)
it becomes more channelized, the exact "transition line" from dispersive to
channelized being a line relating each Dykstra-Parsons value to a critical cor-
relation length.
Interestingly, in our results the ensemble injectivity index reaches both its
minimum and maximum values at highest Dykstra-Parsons coefficient values
and the contour lines for injectivity index form a fan-like pattern. The be-
haviour can best be described such that (i) in the dispersive flow region (lower
right of the plot) injectivity increases with decreasing σ and λ (thereby, with
decreasing macrodispersivity, as it is in this region dispersivity can be defined
in terms of macrodispersivity), and (ii) in the channelized flow region (upper
left of the plot) injectivity increases with decreasing σ and λ (thereby what
can be defined as decreased heterogeneity of the channelized system). In the
channelized flow regime, increases in σ and λ lead to higher effective perme-
ability of the domain, due to the well-connected high permeability channels,
and thereby enhance injectivity. On the other hand, in the more dispersive
flow regime with low correlation lengths, increasing σ causes both high and
low permeabilities to become more extreme, resulting in lower effective per-
meability and worse injectivity (because the effective permeability is locally
governed by the low rather than high permeability values, as in a harmonic
averaging procedure). This general result is also in agreement with previous
study by Lengler et al. (2010) that addresses the issue in the low Dykstra-
Parsons coefficient region.
32
Revisiting the role of the scaling factor ξ in Figure 3.2 and to contem-
plate on its physical meaning, we found that the value of the fitting parameter
ξ corresponds to one correlation length where all the injectivity curves cross
the Iin j value of around 1×10−3. At this correlation length the injectivities are
independent of σ . Below this correlation length, an increasing level of hetero-
geneity (increasing σ ) reduces injectivity and above this correlation length it
increases injectivity, the effect being stronger the more extreme the correla-
tion length (most extreme lowering at small λ , most extreme increase at high
λ ). This in turn could be linked to the discussion of channelized and dis-
persive flow. As discussed above, we can expect injectivity to increase with
increasing σ in the more channelized flow regime and to decrease in the more
dispersive regime. One of the apparent physical explanations to the term ξ , is
that it is essentially a characteristic dimensionless correlation length that dis-
tinguishes systems which are prone to primarily dispersive flow to those prone
to channelized flow.
A similar plot for the storage efficiency coefficient is given in Figure
3.5. The storage efficiency coefficient is seen to systematically decrease with
increasing heterogeneity, that is increasing σ and λ throughout the region, re-
gardless of whether the flow is dispersive or channelized. This is consistent
with the result in Figure 3.3. The effect of increasing σ decreases the pore vol-
ume contacted by CO2 and thereby the storage efficiency, as also depicted in
Figure 3 (in Paper I). It should be pointed out though, that the increasing / de-
creasing effect heterogeneity has on storage depends on the combined effects
of injection geometry and correlation structure of heterogeneity (see Paper I
for more discussions).
3.2 Upscaling of Heterogeneity (Paper II)
Reservoir scale modelling involves different types of media and their het-
erogeneity can exist in multiple scales. In large scale modelling, there is a
limitation in accounting for the details of the multi-scale heterogeneity of the
medium, because the time needed for addressing detailed heterogeneity nu-
merically at a smaller scale is significant.
The idea of upscaling is to use proper averaged properties so that the
numerical simulation can be done using a coarser geological model built for a
large domain. Appropriately averaged medium properties are needed for this.
In the case of CO2 injection, the constitutive relationships for the multi-phase
flow system need to be considered in such upscaling. This issue was addressed
in Paper II, using the data from South Scania site as basis for the analysis.
33
Figure 3.6. (a) Example (log10k permeability field realization for a multimodal hetero-
geneous medium; (b) CO2 saturation distribution obtained from a percolation model
(see the enclosed paper) for an externally applied pressure difference of 20,000 Pa be-
tween the invading CO2 and the formation brine; and (c) CO2 saturation distribution
obtained from TOUGH2/ECO2N simulation.
3.2.1 Multimodal distribution of permeability
In Paper II, a 2D heterogeneous section is considered (40m× 5m) and
composed of a rectangular array of cells. The array is assigned a local perme-
ability that follows a multi-modal distribution. The representative geological
settings are based on the South Scania site where the heterogeneous medium is
characterized by three lithofacies namely sandstone, siltstone and clay (Gun-
narsson, 2011) with the sandstone denoted as the background facies.
The multimodal heterogeneous medium is generated following the method
detailed by Lu and Zhang (2002). First, 100 two dimensional (80×50) Marko-
vian random fields of the three materials are generated with specified propor-
tions and mean lens lengths in both directions using Transitional Probability
Geostatistical Software (T-ProGS) developed by Carle and Fogg (1996, 1997).
Then, for each material, 100 realizations of two-dimensional Gaussian fields
assuming an exponential variogram with log10k∼N(0,σ) are generated using
the random field generator HYDRO_GEN (Bellin and Rubin, 1996). Multi-
modal heterogeneous realizations are then obtained through combining each
Markovian random field with three Gaussian realizations (for the three mate-
rials) that are scaled from zero mean and unit variance to the specified means
and variances. An example realization is shown in Figure 3.6.
Once the multimodal heterogeneous permeability field has been gener-
ated, local scale Pc-S curves are assigned to each cell and scaled according to
its permeability using the Leverett function (Eq. 2.3). 10 cases are selected
for detailed analysis for which 100 realizations are simulated. The sandstone
parameters are based on a representative Berea sandstone sample presented
by Pini et al. (2012). The siltstone parameters are assigned by assuming that
the permeability is smaller than that of the sandstone by about one and a half
orders of magnitude (Kitajima et al., 2005). The clay permeability is set to
be very low (10−17m2) such that the high entry pressure prevents CO2 inva-
34
sion into the clay material. The porosity is assumed to be uniformly 0.2 for
simplicity.
A Brooks-Corey relationship (Brooks and Corey, 1964) is used to repre-
sent the local capillary curves according to
Se = (Pc
Pcd)γ (3.2)
where γ is the local-scale pore size distribution and Se is an effective wetting
phase saturation given by
Se =Sw−Swr
1−Swr(3.3)
where Swr is the local residual wetting phase saturation. The relative perme-
ability for wetting (w) and non-wetting (nw) is given by:
krw = (Se)3+2/γ (3.4)
krnw = (1−Se)2[1− (Se)
1+2/γ ] (3.5)
3.2.2 Upscaling of constitutive relationships
Modelling is carried out with a large-scale average capillary pressure cal-
culated as (Kueper and McWhorter, 1992):
Pc,L =1
Vnw
∫Vnw
PnwdVnw− 1
Vw
∫Vw
PwdVw (3.6)
where Pc,L is the large scale average capillary pressure, Pnw and Vnw are the
local non-wetting phase pressure and the total volume of the non-wetting fluid
and Pw and Vw are the local wetting phase pressure and the total volume of the
wetting fluid. The average non-wetting phase saturation Snw,L is given by
Snw,L =
∫VnwdV
V∫
φdV(3.7)
where V is the total volume of the system and φ is the local porosity; the
average wetting phase saturation is given by
Sw,L =
∫VwdV
V∫
φdV(3.8)
To construct the large-scale capillary pressure curve, the simulation pro-
ceeds by incrementally increasing ΔPex. Each small increment produces a fluid
distribution which gives a (Sw,L, Pc,L) point on the curve. The fluid distribution
associated with each Pc,L also contains the local saturation information which
can be used to calculate the local relative permeabilities krw and krn. Large-
scale average relative permeabilities can be computed through the application
35
Figure 3.7. Effect of sandstone proportion on large-scale capillary pressure curves.
Gray lines show results of 100 realizations for each of the four cases. Black lines
show the respective averages. The local curves for the sandstone (red dashed lines)
and the siltstone (blue dashed lines) are also plotted, with parameters listed in Table
1, Paper II
Figure 3.8. Effect of sandstone proportion on large-scale relative permeability curves.
Gray lines show results of 100 realizations for each of the four cases. Black lines show
the respective averages. Red and blue dashed lines show the local relative permeabil-
ities of the sandstone and siltstone, plotted using Eqs. 3.4 and 3.5 with parameters
listed in Table 1, Paper II
36
Figure 3.9. Comparison of ensemble average capillary pressure and relative perme-
ability curves for different material proportions.
of a single-phase flow-averaging method (Eichel et al., 2005). It is assumed
that the motion of one fluid has no impact on that of the other fluid. Impos-
ing a unit pressure gradient, we solve the pressure equation for a single phase.
We then determine the velocity field from the pressure distribution of the sin-
gle fluid. Subsequently, the effective permeability of the phase considered,
ke f f (associated with a large scale average saturation S), is calculated from the
mean velocity and the applied pressure gradient. The relative permeability is
calculated as the ratio between ke f f (S) and ke f f (S = 1). Repeating this proce-
dure for different Pc,L and thus different S, we obtain the relative permeability
curves for the principle displacement direction.
The large-scale breakthrough capillary pressure (Pb) may be considered
as a characteristic capillary pressure for the large scale. Breakthrough capil-
lary pressure is defined as the externally applied pressure at which the non-
wetting phase connects through the medium. We have obtained the upscaled
absolute permeabilities in the horizontal direction (k) through solving the sin-
gle phase pressure equation. Upscaled constitutive relationships can thereby
be achieved through curve fitting.
Multiple realizations are considered to evaluate the large-scale constitu-
tive relationships. The results are shown in Figure 3.7 for the Pc-S curve and in
Figure 3.8 for the kr-S curve. The ensemble effect of the sandstone proportion
on the large-scale constitutive relationship is shown in Figure 3.9. A more
detailed discussion is given in Paper II.
3.2.3 Upscaled simulations
In order to test the performance of the upscaled Pc and kr relationships
when applied to an upscaled homogeneous model for the domain, we have
carried out simulations of CO2 injection using TOUGH2/ECO2N and com-
pared the migration pattern in a selected heterogeneous model to that of the
corresponding upscaled homogeneous one. A CO2 injection rate of 0.18×
37
10−2kg/s is simulated for both the heterogeneous domain and the homoge-
neous domain. Results for an example realization of the heterogeneous per-
meability fields of Case 2 (Table 1 in Paper II ) are discussed here. The up-
scaled capillary pressure-saturation relationship can be well fitted to the van
Genuchten function (Van Genuchten, 1980)
Pc,L = α[S−1/me,L −1]1/n (3.9)
where Se,L is the large-scale effective brine saturation. Se,L is defined as
Se,L =Sw,L−Swr,L
1−Swr,L(3.10)
where Sw,L is large-scale brine saturation and Swr,L is large-scale residual brine
saturation. The fitting parameters are α = 7680Pa, n = 2.04, m = 1− 1/n =0.509, and Swr,L = 0.302, respectively. The upscaled liquid phase relative per-
meability relationship can be well fitted to the Brooks Corey model given
in Krevor et al. (2012) with the fitting parameter Nw = 8.5. The upscaled
gas phase relative permeability can be fitted to obtain a functional form of
krn,L = 0.7(1− Se,L)3. The upscaled absolute permeability of 2.2× 10−13m2
is used for the homogeneous model. Fig. 3.10 presents a comparison between
the heterogeneous and the upscaled homogeneous model for the CO2 satura-
tion profile (the saturation is averaged in the direction normal to injection).
It can be seen that the upscaled homogeneous model can provide reasonable
results in terms of CO2 migration through the domain, suggesting that the
upscaling method produces a reasonable presentation of this strongly hetero-
geneous system.
Figure 3.10. Comparison of CO2 saturation profile at two different times in the het-
erogeneous model and the homogeneous model with upscaled parameters.
38
3.3 Gaussian Emulator Methods for Modelling aHeterogeneity System (Paper III)
In the previous sections, classical Monte Carlo approaches have been
used for modelling heterogeneous permeability fields. However, the preci-
sion of the results of a Monte Carlo analysis (posterior) depends the ensemble
size, N. Addressing forward propagation of modelling uncertainty (uncer-
tainty analysis, UA) relies heavily on the computational resources. Therefore,
there is great interest in reduced-order models that can capture the essential
behaviour of the (fully) physically based models yet avoiding the prohibitive
computational cost (Razavi et al., 2012). In this section, a new method is
developed to explore the behaviour of the heterogeneous system along with
modelling uncertainty by using a Gaussian process emulator.
3.3.1 Principle of Gaussian process emulation
The objective of UA is to find the distribution of f (X) given the distribu-
tion of X, where f (·) represents the simulator output (either the total mass or
the CO2 breakthrough time). We are interested in estimating the cumulative
distribution functions (CDF):
F(y) = P( f (X)≤ y).
Conventionally we use Monte Carlo simulations if sufficient computer power
is available. If Z1, . . . ,Zn represents a large number of permeability field real-
izations from the log-GRF, then the empirical CDF (ECDF),
F(t) =1
n
n
∑i=1
I f (Zi)≤t , (3.11)
is an unbiased estimator of the CDF. Here, IA is an indicator function taking
value 1 if A occurs and 0 otherwise.
Constructing the ECDF is like fitting a curve; if we can carefully select a
set of {Zi,yi}, we probably can fit a "good enough" ECDF curve using much
less data points comparing to classic MC. We may however leave many gaps
in-between each selected data point along the curve. In fact, we can build a
Gaussian Process emulator to fill the gaps (Pronzato and Müller, 2012). Com-
paring to the expensive simulator the computational cost of running an emula-
tor is negligible.
Gaussian process (shorthand for GP) emulation is similar to kriging. An
emulator (Kennedy et al., 2008) is a statistical model that closely mirrors the
simulator, f (·). A Gaussian process emulator is regarded also as a mathemat-
ical model f (·). Building an emulator is in essence an exercise of regression
in parameter space. The prediction made by an emulator is in the form of
distributions.
39
3.3.2 Parameterization and experiment design
The challenges in adapting the Gaussian processes emulator are first the
smoothness of the modelling output and the dimension of the problem. Con-
sider the conceptual model depicted in Figure 3.1, the heterogeneous model
domain has 100× 20 grid elements, that is, a set of 2,000 values. The initial
RF Z(·) defined on the model domain is location-dependent and therefore has
a very high dimensionality. It is assumed that Z has a spatial correlation struc-
ture specified by a matrix Σ that has an exponential variogram (Eq. 2.1). Thus
if the permeability value at x is known, it is highly informative for x+ ε if εis small. A key to construct a GP emulator (as well as the associated UA) is to
find a lower dimensional representative input that can still capture most spatial
variability in Z. Approaches such as principal component analysis (PCA) have
been used to reduce the problem dimension (Razavi et al., 2012).
If we let Z ∼ N(0,Σ), for some m× n covariance function, we can then
express Z as:
Z(x) =∞
∑i=1
ξiλiφi(x) (3.12)
where the λi are the ordered eigenvalues of Σ, and φi the corresponding eigen-
functions. The ξi are independent N(0,1) random variables. This is a special
case of Karhunen-Loève expansion (Cliffe et al., 2011; Schwab and Todor,
2006) and it decomposes the random process into the sum of independent ran-
dom variables. Note that the expression for Z is exact for a finite dimensional
representation (in our case the Z has a fixed number of 2000 elements). As λiis a decreasing sequence, we can truncate Z to its first d terms using
Z(x) =d
∑i=1
ξiλiφi(x) (3.13)
where Z is a lower dimensional representation of Z.
To reconstruct Z, only ξi needs to be saved since λi and φi are all the
same for the targeted field. This truncation explains the most variance and
achieves the minimum mean square error amongst all such approximations.
We will exploit this truncation to build a reduced order emulator. By building
an emulator from Z rather than Z, which is equivalent to building an emulator
with input ξ1, . . .ξd , we have reduced the dimensionality of the problem from
n to d, where all we require is that f (ξ ) is similar to f (ξ +δ ) when δ is small.
The experiment design is then, to find an optimum space filling ensemble for
ξi ∈Rd which can be done by, such as maxmin Latin hypercube design (Morris
and Mitchell, 1995). To fit a Gaussian process emulator one must further
specify a prior mean and a covariance function. More details are found in
Paper III.
By using GP emulator, the initial UA with regards to {Zi,yi}Ni=1 is now
replaced with {ξi, yi}Mi=1 where M and N are the size of data points and more
40
importantly, M � N. Note that the UA on f has now been translated into a
much lower dimensional one.
3.3.3 Applying Gaussian process emulation for uncertaintyanalysis
The objective in our example is to produce the empirical cumulative dis-
tribution function (ECDF) of breakthrough time (BT) and total CO2 mass
(TM) by using as few simulator runs as possible saving computational cost
compared to Monte Carlo methods. The domain is similar to that in Paper I.
One GP is built using a selected training set {ξi, yi} following the procedure
described in the previous section. For each of the scenarios (Table 3.1 ) a
standalone emulator is created.
We generally follow the procedure suggested by Oakley and O’Hagan
(2002) for CDF approximations:
1. Generate a set of 1000 test points ξξξ ∗i ∼ N{0,I}, i = 1, . . . ,1000 ;
2. Evaluate the emulator at those ξξξ ∗i to get a set of yyy∗i = f (ξξξ ∗i );3. Calculate the quantiles for these yyy∗i and form its corresponding ECDF
by using Eq.3.11.
Repeat the process L times to obtain a family of F∗j=1, . . . ,F∗j=L ECDFs. We
can then calculate the lower, upper, mean or median quantiles for the targeted
ECDF of our interest.
In our case, we use the median ECDF as an approximation to the true
CDF as:
F∗(t) = Median(F∗j (t)), j = 1, . . . ,L(L = 100 in our case) (3.14)
The construction of the Gaussian Process emulator starts from the exper-
iment design, which in our case is in connection with the generation of the RV
Z. A careful design using Latin Hypercube Sampling (McKay et al., 1979) was
conducted with criteria of maximizing the minimum distance (in ξ ) in order
to cover the full range of the aforementioned uncertainty. The design points
used for emulator training consist of far less data points than the Monte Carlo
set. For the three values considered (λ = 0.075,0.15,and 0.30), we generate
800, 400 and 400 design points, respectively (Table 3.1).
The implementation of the Gaussian process emulator is based on GP-
stuff, a set of computer codes integrating Gaussian process models for Bayesian
analysis (Vanhatalo et al., 2012). The GP structure was created by defining the
likelihood and covariance function. We considered Gaussian likelihood and
Matérn covariance function (ν = 3/2). Note that the covariance function (also
called kernel) here should be distinguished from the variogram (Section 2.2.2).
The covariance function used in the GP structure can roughly be thought of as
describing the distance in the high dimensional input space.
41
Monte Carlo set (truncated)
TOUGH2/ECO2N simulator
log-GRF input:
Gaussian Processemulator
Training inputs Real valued targets :
Training using selected
GP Emulator
MC ECDF (true CDF)
MC design
Multiple new inputs :
Median ( ) ~ CDF*
Multiple output ECDF result:
Monte Carlo (MC)
Predicting using GP emulator
Figure 3.11. Comparing the procedure of CDF calculations using the
TOUGH2/ECO2N simulator and the Gaussian process emulator. The thickness of
the arrow illustrates the relative computational cost.
Table 3.1. Case specifications and results for model selectionCase No. 1 2 3
Correlation length (λ ) 0.075 0.15 0.30
size of MC set (N) 10,000 10,000 10,000
size of Training set (M) 800 400 400
number of K-L coefficients (d) 30 20 20
CRPSBT,Martérn 0,00640 0,00193 0,00153
CRPST M,Martérn 0,00490 0,00766 0,00975
CRPSBT,SE (d = 20) 0,00108 0,00187 0,00135
CRPST M,SE (d = 20) 0,02489 0,02508 0,02534
It is assumed that the ECDF generated by the Monte Carlo TOUGH2
simulations is a true CDF which is denoted by F(·) (Fig. 3.11). One GP
emulator ( f (·)) is built for each modelling output of our interest (i.e., BT and
TM). Multiple ECDFs (F∗) are produced by running each GP a large number
of times. Note that we used the median of multiple ECDFs to approximate the
corresponding true CDF.
3.3.4 Comparison to the conventional Monte-Carlo approach
The two-phase flow behaviour as well as the migration pattern of the
CO2 injected into a deep saline aquifer have been discussed previously (Sec-
tion 3.1). We here focused on the assembly of the ensemble behaviour of
the breakthrough time and the total mass accumulated using the GP emulator.
Each quantity of interest (TM or BT) from each of the three correlation length
cases was considered as a standalone model process. As the training set being
based on Latin Hypercube design, we used a fixed number of training points
42
(Table 3.1) to construct each of the three GP structures. For each emulator
run, 1,000 random sample points were first generated using a pseudorandom
number (vector) generator in Matlab assuming a dimension corresponding to
d = 30 (Case 1) or d = 20 (Case 1 and 2). Then this set of random vectors,
altogether with the corresponding training pairs (inputs and training targets),
were used to feed the designated GP structure in order to produce / draw one
sample from the posterior distribution. For each quantity of interest, L = 100
posterior samples were used to calculate the ensemble ECDF.
To evaluate the use of the emulator for empirical CDF prediction, we cal-
culated the Continuous Rank Probability Score (CRPS) (Gneiting and Raftery,
2007):
CRPS(emulator) =1
L
L
∑i=1
∫ x=∞
x=−∞(F∗i(x)−F(x))2dx (3.15)
where F∗i is the posterior sample produced by the GP, F is the empirical CDF
from the MC, i is a counter and L is the total number of the sample drawn
from the GP posterior. Notice that each of the samples can resemble a CDF
function that is discrete and defined in the range of [0,1]. If we find that
CRPS( f1) ≤ CRPS( f2), we can deduce that emulator f1 is superior to f2 for
our purposes. The smaller the CPRS value is, the better can the use of GP
approximate the MC CDF (Table 3.1).
The results from the GP are compared to the ones from classical MC (see
Figure 3.12). The confidence intervals of the MC CDF are omitted deliber-
ately for visual clarity. The dashed lines (posterior credible intervals) indicate
that the MC CDF can be enveloped by using merely 100 emulator runs. Ex-
cellent matches are observed: for all cases examined, the median GP curves
can replicate the MC ones almost exactly.
It has been concluded in Paper III that the GP emulator can be well
adapted to explore the uncertain underground CO2 behaviour caused by per-
meability heterogeneity.
43
log10(t) (seconds)4.5 5 5.5 6 6.5 7 7.5 8
F
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1BT, Case 1
MC CDFGPGP 95% percentile
Total Mass (kg) 1041 1.5 2 2.5 3 3.5 4
F
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1TM, Case 1
MC CDFGPGP 95% percentile
Figure 3.12. Empirical cumulative density functions for the breakthrough time (upper
panel) and the total mass (lower panel). MC results are denoted by black solid lines.
GP results are denoted by red solid lines. The dashed-lines show 95% confidence
intervals for GP results.
44
4. Modelling of Large CO2 Storage Systems
Predictive simulation of GCS requires proper evaluation of storage ca-
pacity as well as modelling of the long-time fate of the injected CO2 at large
scale. The complexity level of such modelling is often limited by computa-
tional resources and data availability which make the most appropriate level
of modelling ambiguous (Celia and Nordbotten, 2009). For a given set of
reservoir properties, both the maximum possible injection pressure that can
be used and the migration distance of the injected CO2 can become limiting
factors. However, which one of the two is the dominant limiting criteria is
unknown a priori (Szulczewski et al., 2014). Reliable, efficient, basin-scale
models both for pressure build-up and for CO2 transport are needed.
As part of site-scale modelling of two industrial scale sites, namely the
South Scania site (Paper IV) and the Dalders Monocline site (Paper V) we
have addressed this issue and developed an approach where models of increas-
ing complexity are used to determine the sites’ capacity to store CO2. In the
following text the principle of such modelling approach is described along
with its application for the study of these two aforementioned sites.
4.1 Large Scale Models of Varying ComplexityThe principle of the integreated modelling approach with models of in-
creasing complexity being successively used is summarized in Figure 4.1. As
the first step a semi-analytical model based on the models by Mathias et al.
(2009) is employed. The objective of this model is to provide an order-of-
magnitude first estimate of the maximum injection rates that can be used with-
out exceeding the maximum allowable pressure that can be employed without
causing any damage to the rock, thereby compromising the integrity of the
storage. Based on the results of this model, we then proceed to more detailed
numerical models that allow a more detailed accounting of medium properties
and boundary conditions, first based on the vertical equilibrium (VE) approach
(Bear, 1972) as developed to CO2 storage by e.g. Nordbotten et al. (2005)
and Gasda et al. (2009). The VE is still simplified in comparison to a full-
physics model in three dimensions, but gives a better estimation of the plume
spreading than the semi-analytical approach. Finally based on the results of
the modelling with the VE model the scenarios for a full three-dimensional
numerical model TOUGH2 of the site are selected to allow a detailed analysis
of site behaviour under CO2 injection and storage.
A brief summary of the main procedure is summarized as follow:
45
1. The semi-analytical solution following (Mathias et al., 2009) is first
used for a wide range of injection rates to investigate the injectivity
constraints and the parameter sensitivity under different boundary
conditions and varying parameter values.
2. For all the viable injection rates identified in (1), we then use the VE
model (Gasda et al., 2009; Nordbotten et al., 2005) to investigate the
migration limits, and to select an injection rate or range of injection
rates that can meet both injectivity and migration constraints.
3. Finally, full three-dimensional simulations are performed using
TOUGH2 / TOUGH2MP simulator for the selected injection rates
(based on (1) and (2)) and to get a detailed picture of the CO2 spread-
ing and the related pressure effects. The results of the three ap-
proaches are compared throughout the analyses.
The difference approaches are briefly explained below:
Figure 4.1. Order of analyses and modeling methods used (Paper IV and V).
The semi-analytical model (Paper IV,V and Mathias et al., 2011a,b) is
used for pressure buildup estimation. The solution is obtained through analyt-
ically solving the radially symmetric, two-phase two-component flow equa-
tions (Mathias et al., 2011a,b). The solution assumes a horizontal formation,
impermeable caprock and baserock, vertical pressure equilibrium, homoge-
neous fluid and formation properties and negligible capillary pressure effect.
The solution takes into account the effect of CO2 compressibility by iteratively
calculating the CO2 density based on the actual near-well pressure buildup.
While the standard solution assumes open boundaries, the model can also take
into account the effect of impervious fault boundaries by including superposi-
46
tion of pressure response by means of the method of image wells (Yang et al.,
2015). The semi-analytical solution is described in more detail in Paper IV.
The vertical equilibrium approach is a sort of quasi-3D modeling tech-
nique, assuming equilibrium of pressure in the vertical direction (Bear, 1972).
This numerical scheme was originally developed to predict regional ground-
water movements in unconfined aquifers, but has later been extensively used
by petroleum industry due to its accuracy and computational simplicity (Gray
et al., 2012). Recently, this method has been spotlighted again and used for
CO2 storage modeling, based on the similarity of physical properties of su-
percritical CO2 and liquid petroleum under certain conditions (Gasda et al.,
2009; Juanes et al., 2009). The details of the VE formulation as applied to
GCS are found in Gasda et al. (2009). Here the Implicit Pressure Explicit
Saturation (IMPES) scheme is used, and the implicit standard finite element
method (FEM) is adopted for solving pressure and the explicit control volume
FEM for saturation, to ensure mass conservation and avoid numerical difficul-
ties. The capillary pressure between phases and the dissolution of CO2 are
not considered in the work of this thesis. The standard one-layer version of
the VE model is applied in Paper V. The VE model was extended to enable
analysis of multi-layer system feature by coupling multiple VE models with
one-dimensional Darcy flow. The detailed implementation of this is given in
Paper IV.
The TOUGH2 / TOUGH2MP model is the most comprehensive simu-
lator in the scheme of large-scale GCS modelling analyses. TOUGH2MP is
a massive parallel version of the established TOUGH2 code (Pruess et al.,
1999), which is designed for the most computational demanding 3D reservoir
simulations (Zhang et al., 2008). By integrating the equation of state module
ECO2N (Pruess and Spycher, 2007), T2MP/ECO2N is capable of resolving
the underground CO2 migration behaviour in great detail.
4.2 The Migration-limited System — South Scania Site(Paper IV)
The South Scania site is presented in Figure 2.4. The objective of the site
modelling is to give site-specific predictions concerning CO2 storage capacity.
The modelling domain consists of a continuous sequence of the forma-
tion layers R1, C3, R2, R3 and R4, and is considered to be sealed from the top
and the bottom (Fig. 2.5). Any possible leakage and pressure relief through
the caprock and baserock is thereby neglected, which leads to possibly over-
estimating the pressure build-up. The estimate is, however, conservative and
therefore motivated in this preliminary evaluation of the reservoir’s suitability
for CO2 storage. The medium properties used in the modeling are presented in
Table 4.1. It can be seen that the relatively thin reservoir unit R2 has an order-
of-magnitude higher permeability than the other reservoir units, which have
47
Table 4.1. Database parameters for CO2 injection simulationUnit R1 C3 R2 R3 R4
Thickness M 40 32 14 138 40
Permeability (horizontal) mD 88 0.002 1800 160 200
Permeability anisotropy
(horizontal to vertical)
- 1.5 1 3.5 7.5 3.5
Porosity % 23 15 25 20 23
Salinity NaCl g/l 125 125 125 125 125
Pore Compressibility 10−10Pa−1 4.50 - 4.50 4.50 4.50
Brine compressibility 10−10Pa−1 3.54 - 3.54 3.54 3.54
Temperature ◦C 55 55 55 55 55
Mean slope* deg 0.86 0.83 0.81 0.74 0.66
Note *:calculated along a horizontal straight line from FFC-1 to Kungstorp-1 (Ku-1)
permeabilities that are commonly considered acceptable for CO2 storage. In
all of the model analyses discussed below we assume CO2 injection through
the existing deep well FFC-1 for which extensive data are available.
The stepwise approach adapted to South Scania site is as described above:
(i) first the semi-analytical solution is used to screen the possible injection
rates that meet the pressure threshold criteria, (ii)then the VE model is used to
select the injection rate based on migration limits, and (iii) finally TOUGH2
model allowing greatest complexity (3D model, inclusion of solubility and
capillary trapping) is used for the most detailed predictions of the long term
CO2 evolution and detailed inventory for the selected injection scenario.
4.2.1 Injectivity analysis: the selection of CO2 injection ratebased on maximum allowed pressure
The pressure build-up threshold is chosen to be 50% of the initial hydro-
static pressure according to literature (Rutqvist et al., 2007) which corresponds
to Fop = 50% as defined in Eq. 2.6. This threshold describes the pressure that
can be used without compromising the integrity of the caprock. The semi-
analytical approach is first used to establish CO2 injection rate sensitivity to
the pressure build-up according to each of the three boundary condition sce-
narios (BS1 - BS3, referring to Table 3 in Paper IV).
The maximum CO2 injection rates for the different possible boundary
condition scenarios calculated using the semi-analytical approach are sum-
marized in Table 4.2 (first three columns). It can be seen that the effect of
the chosen boundary condition is most significant in R2 where the change
from BS1 (all fault zones closed) to BS2 (all fault zones open) allows for a
doubled injection rate. For the more detailed analysis with the VE model,
the boundary condition scenario BS3 (the most realistic scenario where only
the Romeleåsen Fault is considered closed, see Figure 2.4) was chosen. VE
48
Table 4.2. Estimated maximum CO2 injection rates (Mt / year) for different boundaryscenarios.
Reservoir units BS1 BS2 BS3 Used in numerical simulations∗R1 0.75 1.2 1.1 0.17
R2 2.5 >6 6 1.23
R3 4.0 >6 6 1.03
R4 1.5 2.5 2.2 0.60
Total 7.75 >15.7 15.3 3.00
Note*: based on preliminary modelling with VE model assuming the same distribution
of flow as obtained with semi-analytical model
model is then used to determine the viable injection rate that can sustain both
the maximum allowed injection pressure and the maximum allowed migration
distance. It can be seen that already an injection rate of 3 Mt CO2 per year over
50 years can lead to a notable CO2 flow distance of 35 km in 750 years, where
the plume front in R2 eventually reaches the Öresund and Svedala faults (as
shown in Figure 7 in Paper IV). This relatively fast migration indicates that
the migration is likely a limiting factor at the South Scania site. Based on this,
the injection rates for the follow up simulations in both VE and T2MP were
selected lower than dictated by pressure criteria (first 3 columns in Table 4.2)
and taken as shown in the last column in Table 4.2.
A detailed comparison between VE and T2MP is given in Paper IV. In
terms of pressure prediction at the end of injection, overpressure in R2 is given
respectively as ∼ 8% by VE and about 7% by T2MP while both agree with
the semi-analytical results (∼ 10%).
4.2.2 Migration analysis
Both the VE model and T2MP models provide information on plume mi-
gration, the latter providing a more detailed estimate. It has been identified in
Paper IV that the dominant CO2 spreading is in unit R2 which is a thin layer
but with highest permeability. The CO2 migration is in the up-dip direction.
Both predictions from the VE and T2MP models give similar pattern. The CO2
saturation snapshots in R2 at selected times are given in Figure 4.2 as deter-
mined by T2MP model. The end-of-simulation saturation pattern agrees with
the caprock topography which is an indication of structural trapping. Since C3
has a very low permeability that can work as a perfect seal, the effect of the
caprock in T2MP is effectively equivalent to the confined aquifer assumption
made in VE.
The VE model can produce reasonably good result for the CO2 footprint
despite the fact that it does not include residual and dissolution trapping. The
modelling framework (Fig.4.1) where models of increasing complexity are
successively employed. From Figure 7 in Paper IV we can see that the plume
49
Figure 4.2. CO2 saturation in R2 from T2MP (showing the top of R2). (A) t = 350
years; (B) t = 750 years; (C) t = 1000 years. (D) R2 top elevation.
as calculated with VE model has advanced 35 km in 750 years, which is more
than the prediction by the T2MP model. The model does nevertheless give a
good first estimate that enables guiding the more detailed T2MP simulations.
Overall, the modelling framework (Fig.4.1) where models of increasing
complexity are employed was successfully implemented in the preliminary
capacity analysis of South Scania site, providing information of the limiting
criteria. Based on this, recommendations of better use of the total capacity
of the site include, for example, injecting through horizontal or vertical wells
into the lower units (R3 and R4) only, where the permeability is not as high
but the total thickness considerable.
4.3 The Injectivity-limited System — DaldersMonocline (Paper V)
Dalders Monocline is a prospect structure in Baltic Sea basin that has
been identified to have potential (Cambrian) (Fig. 4.3) for GCS (O’Neill et
al., 2014). The structure extends north west to Gotland in Sweden and cross
several countries either on-shore or off-shore (Fig. 4.4). Its northern boundary
is controlled by the limit of the Faludden Sandstone in Sweden and the Middle
Cambrian in Poland and Estonia. The southern boundary is controlled by the
major faults to the north of the Liepaja Saldus Ridge.
In term of the sealing unit, the Alum Shale does not cover the whole
Dalders Monocline and only overlies the reservoir in the southernmost half of
the domain, as indicated in Figure 4.4 by the curves (A-B’-C-A) which is the
delineated modelling domain, that is, the best caprock properties according to
the present geological knowledge.
50
Figure 4.3. The Dalders Monocline cross-section.
Figure 4.4. Map of the Baltic Sea Basin.
A more detailed geological description is given in Paper V. The geohy-
drological properties used for numerical simulations are shown in Figure 4.5.
The boundary conditions are shown in Figure 4.7a.
4.3.1 Injectivity analysis
A parametric sensitivity study is conducted by using the semianalytical
approach that can set reference values for screening suitable injection locations
given detailed hydrogeological information (permeability, porosity, aquifer
thickness, distance from the injection well to the closed boundary, etc.).
51
Figure 4.5. Depth map of the top of layer (top left), permeability map (top right) and
thickness map (bottom) used for the numerical simulations. The maps are obtained
using the currently available sparse well log data and geologic information.
Depending on the hydraulic diffusivity, the pressure plume travels much
faster and reaches much farther than the CO2 plume. The no-flow boundary
representing an impermeable fault zone has an effect on the pressure buildup.
Figure 4.6 shows the pressure - injection rate response curves for four
Lw−b scenarios (Lw−b is the distance from the injection well to the closed
boundary) at t = 50 years with fixed k = 80 mD and φ = 0.15. The closer
the injection well is to the closed boundary, the greater the pressure increase
becomes.
The current selection of injection wells is desirable to be located not too
far from the southeast no-flow boundary where the formation is deeper, and
preferably coincide with existing well locations (Fig. 4.7a). The permeabil-
ity at injection point A, B, and C is respectively 30, 40 and 80 mD, which
is small in comparison to the case of the South Scania site (Table 4.1). A
maximum injection rate estimated from the semi-analytical solution (assum-
ing 50% maximum allowable pressure increase) is about 0.5 Mt/yr when the
permeability is 50 mD. We use 0.5 Mt/yr for wells A and B and 0.2 Mt/yr for
well C. Note that well C is located in a narrow ’deadend’ and would not be a
suitable location for injection. The purpose of selecting well C is to illustrate
the pressure effect of closed boundaries nearby.
Figure 4.7 presents the pressure buildup in the formation after 50 years
of CO2 injection. Generally, the VE model and the T2MP model show good
agreement on the prediction of overpressure at the injection wells. The VE
model tends to predict slightly higher pressure and a larger pressure plume
than the T2MP model, due to the lack of dissolution processes in our current
version of the code. The overpressure for wells A and B predicted by T2MP
52
Figure 4.6. Effect of the distance from the injection well to the no-flow boundary
(Lw−b) on pressure increase at the injection point. The black dashed line shows the
threshold pressure of 60 bars.
is 43% and 59% of the initial pressure, respectively. These numbers are close
to the rough estimates by the semi-analytical solution.
4.3.2 CO2 plume migration and capacity limiting criteria
The predicted CO2 plumes at injection wells A and B at 50 years have a
radius of a few kilometers for both the VE and the T2MP models (Fig. 4.7c).
Post-injection migration until 500 years has also been simulated, however,
because the plume migration is less than 5 km for the given parameter set (not
visually distinguishable in Fig. 4.7c) it is not shown here.
As the CO2 migration will be dominated by the sliding motion along the
slope at large times, we consider a 2D scenario for investigating the potential
for CO2 up-dip migration together with a 3D case of high resolution simulation
to validate the 2D estimates. In the 2D simulations, we vary two important
parameters controlling the long-term CO2 migration: formation permeability
(k = 30, 100 and 300 mD) and residual gas saturation and (Sgr = 0.1, 0.2 and
0.3). Effects of permeability and residual gas saturation on CO2 plume tip
migration distance is shown in Figure 4.8.
The 3D simulation shows a similar trend as observed in the correspond-
ing 2D case. That is after about 4000 years, trapped CO2 mass (dissolution
plus residual trapping) accounts for 97% of the injected mass. At this time,
essentially all CO2 mass becomes trapped securely in the aquifer by residual
and dissolution trapping. More detailed discussion is referred to Paper V
53
A
B
C
(%) (%)
CO2Saturation
0.600.540.480.420.360.300.240.180.120.060.00
75.0068.5062.0055.5049.0042.5036.0029.5023.0016.5010.00
(a)
(b)
(c)
CO2
Figure 4.7. (a) Boundary conditions and injection well locations. The injection rate
for A, B and C is 0.5, 0.5 and 0.2 Mt/year, respectively. (b) Overpressure distribution
in percentage (Fop) at 50 years; left VE model and right 3D T2MP model. (c) Depth
averaged saturation distribution at 50 years; left VE model and right 3D T2MP model.
54
Figure 4.8. Plume tip migration distance as a function of time with varying k and Sgr.
Under the current injection scenario, the dominant constraint for the CO2
storage potential is the pressure buildup. An injectivity limited capacity is
estimated to be about 100 Mt for a 50-year injection duration. It is unlikely
for CO2 to leak through the north end of the formation and the safe long-
term CO2 containment in the mildly sloping formation can be ensured due to
the residual and dissolution trapping mechanisms. That said, we emphasize
the need for more field data (field measurements of permeability, residual gas
saturation, etc.) to reduce uncertainty in the modelling.
55
5. Summary and Conclusions
This thesis explores methodologies for characterizing deep saline aquifers
for long term storage of CO2. Key research questions include effects of ge-
ological heterogeneity on CO2 storage performance and the predictive mod-
elling of large CO2 geological storage systems. The main findings and con-
clusions are summarized below
In Paper I, the effect of varying heterogeneity characteristics of perme-
ability fields (as described through correlation length λ and standard devia-
tion σ ) on storage performance is explored. Multiple-realization Monte Carlo
simulations are focused on establishing relationships to explain ensemble be-
haviour of storage performance measures by means of simple parameter groups.
The results show that the dependence of injectivity on both λ and σ is well
captured with a linear correlation between injectivity index and a parameter
group (λ/ξ )σ , where ξ is a dimensionless scaling parameter, and the injectiv-
ity index increases with increasing (λ/ξ )σ . The storage efficiency coefficient,
on the other hand, decreases with both increasing λ and σ , and a simple linear
fit is found between E and the parameter group λσ 2.
Paper II looks at multimodal heterogeneity fields and explores a method-
ology of using multiple detailed simulations to derive equivalent properties
at coarser scale for applications in larger scale models. The large-scale con-
stitutive relationships are found mainly to be controlled by the proportion of
background material and its permeability variability, while the existence of the
non-framework materials and their permeability variabilities may contribute,
in a complex way, to the uncertainty in the large-scale constitutive relation-
ships. In addition, the Leverett equation may well describe the relationship
between the large-scale capillary pressure and absolute permeability when the
sandstone (background material) proportion is high (>0.7). For cases with
smaller sandstone proportions the results indicate that it may not be appropri-
ate to link capillary pressure and absolute permeability through the Leverett
equation.
In Paper III, a novel approach of using a Gaussian process emulator
(GPE) is developed and used to explore the forward propagation of uncer-
tainty in the heterogeneous medium permeability through modelling. We re-
visit the model used in Paper I and estimate the cumulative distribution func-
tions (CDF) of the CO2 breakthrough time (to a spill point) and the total mass
using a computationally expensive Monte Carlo (MC) simulation. We then
show that we can accurately reproduce these CDF estimates with the emula-
tor, but using only a fraction of the computational cost compared to the cor-
responding Monte Carlo simulations. In order to build a GPE that can predict
56
the simulator output from a permeability field consisting of 1000s of values,
we use a truncated Karhunen-Loève expansion of the permeability field, and
then use a Bayesian functional regression approach. An individual GPE is
created for each modelling output of interest by considering its explicit mean
and Matérn covariance function (ν = 3/2). For all cases examined, excellent
matches are observed using merely 100 emulator runs.
Paper IV and V focus on the predictive modelling of the CO2 storage po-
tential of large systems namely the South Scania site, Sweden and the Dalders
Monocline in the Baltic Sea basin. In order to maximize the confidence in the
model predictions, the work employs a set of different modelling approaches
of varying complexity, including a semi-analytical model, a sharp-interface
vertical equilibrium (VE) model and a TOUGH2MP / ECO2N model. The
semi-analytical model provides fast estimate of the pressure buildup for a
variety of injection rates as well as the parameter sensitivity under different
boundary conditions or reservoir properties. Given a certain pressure thresh-
old, maximum injection rates are estimated from the semi-analytical model
and are then fed into the VE model allowing more detailed accounting of CO2
migration patterns and reservoir characteristics. The selected injection sce-
nario is thereafter concluded by the most comprehensive T2MP model. From
the analyses, the pressure buildup predicted by the two numerical models fall
close to that by the semi-analytical solution. An agreement in plume migration
patterns is also identified for the VE and T2MP models. It is shown that the
South Scania site under the considered injection scheme is a migration-limited
system whereas the Dalders Monocline is injectivity-limited. Extensive mod-
elling of the respective post-injection migration and trapping evolutions pro-
vide quantitative predictions of the CO2 storage potential.
To summarize, the various topics / methodologies elucidated in this thesis
have been following two main considerations, one is geological heterogeneity
and the other is the problem of scale, both issues posing significant challenges
when real sites are to be modelled for geological storage of CO2. In this the-
sis, methods and models have been developed to aid such analyses. The results
in Paper I demonstrate the overall effects of heterogeneity on storage perfor-
mance and the results can be used to get estimates of injectivity and storage
efficiency if heterogeneity characteristics are known. Paper II, in turn, shows
an approach of actual upscaling of smaller-scale two-phase flow properties to
properties to be used in large scale models. Paper III further presents a totally
new, computationally more effective approach of modelling a heterogeneous
medium that can also be used in other applications. Papers IV and V finally
provide a framework for modelling large scale systems, as well as provide
preliminary information of CO2 storage possibilities related to two large sites.
Some future perspectives for continued research could address especially the
following points;
57
• In terms of the relationships found between heterogeneity character-
istics and injectivity and effective storage coefficient, it is of interest
to explore the form of such relation in other flow and injection ge-
ometries as well, including three-dimensional systems and injection
from horizontal wells.
• In this, the use of Gaussian process emulators may be considered
as well, as it would provide significant computational benefits. Its
Bayesian formulism can also be extended as a sensitivity tool to iden-
tify the parameters that are most important to the modelling outputs
of interests.
• In terms of modelling the geological heterogeneity, the Gaussian pro-
cess emulator can be used in resolving various types of uncertainty,
such as the facies composition and parameters used for constitutional
relationships.
• In terms of modelling the large field sites, a natural next step will
be analyzing the effect of different injection geometries, such as in-
jection from horizontal wells. It is also of interest to see how the
inter-layer heterogeneity would influence the results in these forma-
tions.
58
6. Acknowledgement
I am in debt to Professor Andrew Cliffe at the School of MathematicalSciences, University of Nottingham, who introduced me to the methodologyof Gaussian process emulation. Andrew was diagnosed cancer soon after thestarting of our collaboration on Paper III. He continued to work using scat-tered time while receiving his treatments but unfortunately passed away. Hisinvaluable contribution as well as his passion for research is memorized.
The work has been supported by the European Community’s Seventh Frame-
work Programme (FP7) MUSTANG (No. 227286), PANACEA (No. 282900)
and TRUST (No. 309067) projects. The computations were performed on
resources at Uppsala Multidisciplinary Center for Advanced Computational
Science (UPPMAX), Chalmers Centre for Computational Science and En-
gineering (C3SE) both provided by the Swedish National Infrastructure for
Computing (SNIC), with project numbers: SNIC 2015/1-255, SNIC 2015/6-
128 and SNIC 2016/2-7.
I would like to thank my main supervisor Auli Niemi for her undoubted sup-
port during my PhD study. She has been patient and very carefully keeping me
on track of my research. Her thoughtful advices and timely inputs have guided
me through some very difficult time. I am also very grateful to my assistant
supervisor Fritjof Fagerlund who has offered quick response and valuable in-
structions. He is seemingly more of a marathon champion than a scientist, for
his many, effortless, jaw-dropping marathon records; and he is a continuous
source of inspiration. I am also lucky to have Zhibing Yang as my mentor who
has helped me generously on various occasions, especially with my writing.
He is a friend, and my role model of a scientist.
I am honored to work in the MUSTANG partnership, collaborating with
the great minds shaping up the GCS research: Jacob Bear, Jesus Carrera, Chin-
Fu Tsang, Christopher Juhlin, Mikael Erlström, Jacob Bensabat, Henry Power,
Quanlin Zhou, Manfred W. Wuttke, Stefan Finsterle, Richard Wilkinson, Vic-
tor Vilarrasa, Kenny Zhang, Andrea Borgia and Barry Freifeld. I remember
vividly many of those interesting conversations that have made profound im-
pact in my life.
Throughout my work, I have learned greatly from and been helped by fel-
low researchers and staffs at the Department of Earth Sciences, Uppsala Uni-
versity: Bruno, Farzad, Fengjiao, Lebing, Kristina, Maria, Maryeh, Tomas,
Claudia, Reinert, Beatriz, Albin, Lichuan, Mathias, Kaycee, Erik, Petra, Olof,
59
Johan, Jennie, Christian, Saba, Agnes, Diana, Monica, Adam, Marc, Carmen,
Nino, Sergey, Benoît, Eduardo, Hanna, Tom, Johanna, Dorothée, Ward, Babis,
Cici, Björn, Aubrey, Jean-Marc, Hongling, Fei, Ping, Steffi, Tito, Yongmei,
Fredirik, Eva, Simon, Leif, Cristina, Taher, Zara ... if only I can list all the
names. Tomas Nord is thanked for many times reviving my hard drive. Nina
Svensson is thanked for listening to many of my ideas and theories. Special
thanks to the senior scientists sharing knowledge beyond scientific researches,
Ala Aldahan, Sven Halldin, Allan Rodhe, Keith Beven, Hemin Koyi, Yvonne
Tsang, Roger Herbert, Conny Larsson, and Giuliano Di Baldassarre.
I would like to thank my friends outside the department for sharing many
wonderful moments: Anna Kauffeldt, Ning Zhang, Xia Shen, Kaweng Ieong,
Byeongju Jung, Fuguo Tong, Jiajun Chen, Elias Urquia, Prabhakar Shaman,
José-Luis Guerrero, Peter H. Dimberg, Estuardo Guinea, Ming Zhao, Solomon
Gebrehiwot, Jose Morales, Eva Podgrajsek, Wei Deng, Biao Wang, Ida West-
erberg, Peng Yi, Liang Dai, Bo Xu, Jianxin Wei, Can Yang, Chunling Shan,
Peng He, Haozhou Wang, Jianliang Wang, Julia Hytteborn, Chengjun Wu,
Qingyue Ren, Jinzhi Hu, Lin Jiang, Weiwei Tang, Yanyan Xu, Tong Liu,
Li Li, Hui Liu, Reiko Akiyama, Johan Dahl, Janice Lind, Kaj Kallioinen,
Xingwu Zhou, Ilona Flis, Baotian Wang, Hao Wang, Yao Ge, Tian Wu, Wei
Liu, Jonas Larsson, Wing Cheng, Erik J. Boström, Xiaoqing Tu, Xiao Wang,
Dafei Huang, Michael van der Meer, Fengping Wu, Mikael Carlsson, Weiping
Zhang and Michael Gustafsson.
I would like to acknowledge the International Finance Corporation (IFC)
for funding my salary for the last few months, in particular to my friend Fran-
cisco Marcos Avendano, Olga Khlebinskaya and Dr. Stephen Alan Hammer,
for their continued support. My gratitude also goes to the Mentor4Researchprogram, in particular to Moa Fransson, Nhils Forslund and Per Kjellin at
UU Innovation for funding and advisory support, and to Johannes Sandberg, a
wonderful scientist, entrepreneur and mentor, who eventually taking me on an
new journey towards the commercialization of my research idea.
Thanks to my sister Fang Tian ( ) in Beijing for her unconditional sup-
port of my pursuing of my career as well as taking good care of my parents.
Also thanks to my parents for their love.
Finally, thanks to my wife, Caihong ( ), this work is as much yours as
it is mine. And also to Joachim ( ), my five-year-old, thanks for the paint-
ing on the Thesis cover, through your eyes I see the beauty of heterogeneity;
and to Xiuchang ( ), my two-and-a-half-year-old, your toddling around is
just to show me that the uncertainty is the nature of everything.
60
7. Sammanfattning på svenska
Denna avhandling undersöker metoder för att karakterisera djupa salt-
vattensakviferer för lagring av koldioxid över lång tid. Särskilda forsknings-
frågor inkluderar effekter av geologisk heterogenitet på förmågan hos geolo-
giska formationer att lagra koldioxid och prediktiv modellering av stora ge-
ologiska koldioxidlagringssystem. De viktigaste resultaten och slutsatserna
sammanfattas nedan. I Artikel I utforskas effekten av varierande heterogen-
itetsegenskaper hos permeabilitetsfält (som beskrivs genom korrelationslängd
λ och standardavvikelse σ ) på lagringsprestanda. Monte Carlo-simuleringar
med multipla realiseringar av permeabilitetsfä ltet syftar till att ta fram rela-
tioner som kan fö rklara ensemble-beteendet hos lagringsprestanda-parametrar
med hjälp av enkla parametergrupper. Resultaten visar att beroendet av injek-
tivitet på både λ och σ representeras väl med en linjär korrelation mellan in-
jektivitetsindex och en parametergruppen (λ/ξ )σ , där ξ är en dimensionslös
skalningsparameter. Injektivitetsindex ökar med ökande (λ/ξ )σ . Lagringsef-
fektivitetskoefficienten, åandra sidan, minskar med både ökande λ och σ , och
en enkel linjär anpassning hittas mellan E och parametergruppen λσ2.
Artikel II undersöker multimodala heterogenitetsfält och utforskar en metod
att använda flera detaljerade simuleringar för att härleda motsvarande egen-
skaper på grövre skala, för applikationer i mer storskaliga modeller. De storskaliga
konstitutiva relationerna visar sig främst styras av proportionen av bakgrunds-
material (grundstruktursmaterial) och dess permeabilitetsvariationer, medan
förekomsten av andra material (ej grundstruktur) och deras permeabilitetsvari-
abilitet kan bidra på ett komplext sätt till osäkerheten i de storskaliga konstitu-
tiva relationerna. Utöver detta kan Leverett-ekvationen väl beskriva förhållan-
det mellan det storskaliga kapillärtrycket och den absoluta permeabiliteten när
sandstenshalten (bakgrundsmaterial) är hög (> 0,7). För fall med mindre sand-
stenshalter tyder resultaten på att det inte är lämpligt att koppla kapillärtryck
och absolut permeabilitet genom Leverett-ekvationen.
I Artikel III, utvecklas en ny metod för användning av en Gaussisk process-
emulator (GPE), och denna används för att utforska framåt-felfortplantning
pga osäkerhet i permeabiliteten hos det heterogena mediet genom modellering.
Vi återvänder till den modell som används i Artikel I och uppskattar kumula-
tiva fördelningsfunktioner (CDF) för koldioxidfasens transporttid genom sys-
temet (genombrottstid) och den totala massan med hjälp av en Monte Carlo-
simulering (MC) som kräver stor beräkningskraft. Vi visar dåatt vi exakt kan
återge dessa CDF-uppskattningar med emulatorn, men med bara en bråkdel av
beräkningskostnaden jämfört med Monte Carlo-simuleringen. För att bygga
61
en GPE som kan förutsäga simulator-resultaten från ett permeabilitetsfält bestående
av 1000-tals värden, använder vi en trunkerad Karhunen-Loève-expansion av
permeabilitetsfältet, och sedan används en Bayesiansk funktionell regression-
smetod. En individuell GPE skapas för varje modellerings-utdata av intresse
genom att betrakta explicita medelvärden och Matérn-kovariansfunktionen (ν =3/2). För samtliga undersökta fall uppnåddes utmärkt passning med endast
100 emulator-simuleringar.
Artikel IV och V fokuserar påprognosmodellering av koldioxidlagringspo-
tential hos stora system, vilka är Södra Skåne, Sverige och Dalders Mono-
cline i östersjön. För att maximera tillförlitligheten hos modellprediktionerna,
används en uppsättning olika modelleringsmetoder med varierande komplex-
itet, inklusive en semi-analytisk modell, en vertikal jämviktsmodell (VE) med
skarpt gränssnitt och en TOUGH2MP/ECO2N-modell. Den semi-analytiska
modellen ger snabb uppskattning av tryckökningen för en mängd olika injek-
teringskvoter samt av parameterkänslighet vid olika randvillkor eller reser-
voaregenskaper. Med ett givet tröskelvärde för trycket uppskattas maximala
injektionshastigheter med den semi-analytiska beräkningsmodellen och matas
sedan in i VE-modellen, vilket tillåter en mer detaljerad uppskattning av koldiox-
idens migrationsmönster och av reservoarens lagringsegenskaper. Det valda
injekteringsscenariot väljs därefter utifrån den mest omfattande T2MP-modellen.
Analyserna visar att den tryckökning som förutsägs av de två numeriska mod-
ellerna faller nära resultatet från den semianalytiska modellen. överensstäm-
melse i plymens migrationsmönster ses ocksåmellan VE och T2MP-modellerna.
Studien visar att Södra Skåne är ett migrerings-begränsat system, medan Dalders
Monocline är ett injektivitets-begränsat system. Omfattande modellering av
hur koldioxidens migration och fastläggning utvecklas över tid i respektive
system ger kvantitativa prediktioner av formationernas koldioxidlagringspo-
tential.
För att sammanfatta; de olika ämnen/ metoder som belysts i denna avhan-
dling har följt tvåhuvudsakliga frågeställningar, den första är geologisk het-
erogenitet och den andra är skalningsproblemet. Båda dessa frågor utgör
stora utmaningar när verkliga områden ska modelleras för geologisk lagring
av koldioxid. I denna avhandling har metoder och modeller utvecklats för
att underlätta sådana analyser. Resultaten i Artikel I visar de samlade ef-
fekterna av heterogenitet pålagringsprestanda och resultaten kan användas för
att fåuppskattningar av injektivitet och lagringseffektivitet om heterogenitet-
segenskaperna är kända. Artikel II i sin tur visar en metod för uppskalning av
egenskaperna hos småskaliga tvåfasflöden till egenskaper som kan användas
i storskaliga modeller. Artikel III presenterar vidare en helt ny, mer beräkn-
ingseffektiv metod för modellering av ett heterogent medium, som kan använ-
das även i andra applikationer. Artikel IV och V ger slutligen ett ramverk
för modellering av storskaliga system, samt preliminär information om la-
gringsmöjligheter av koldioxid i två stora potentiella lagringsområden.
62
Några framtidsperspektiv för fortsatt forskning skulle framför allt kunna
ta upp följande punkter;
• När det gäller relationerna som finns mellan heterogenitetsegenskaper
och injektivitet och effektiv lagringskoefficient är det av intresse att
undersöka hur sådana relationer skulle se ut ocksåi andra flödes- och
injektionsgeometrier, bland annat i tre-dimensionella system och vid
injektion från horisontella brunnar.
• För att göra detta kan en Gaussiska process-emulator användas, efter-
som det skulle ge betydande beräkningsfördelar. Dess Bayesiska for-
malism kan ocksåvidareutvecklas för känslighetsanalyser för att iden-
tifiera de parametrar som är viktigast för modellutdata.
• När det gäller modellering av geologisk heterogenitet, kan den Gaus-
siska process-emulatorn användas för att hantera olika typer av os-
äkerhet, såsom sammansättning av bergarter och parametrar som an-
vänds för konstitutionella förhållanden.
• När det gäller modellering av stora områden, kommer ett naturligt
nästa steg att vara analys av effekten av olika injektions-geometrier,
såsom injektion från horisontella brunnar. Det är ocksåintressant att
se hur heterogenitet i mellanliggande skikt skulle påverka resultaten i
dessa formationer.
63
8.
I
λ σ
(λ/ξ )σ
ξ(λ/ξ )σ
λσ 2 λσ2 ,
II
> 0.7 Leverett
Leverett
III GPE
I
CDF
1000
Karhunen-Loève
Matérn = 3/2
64
100
IV V
Dalders
VE TOUGH2MP / ECO2N T2MP
VE
T2MP
VE T2MP
Dalders
I
II
III
IV V
:
•I
• I III
III
•II
•IV V
65
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Acta Universitatis UpsaliensisDigital Comprehensive Summaries of Uppsala Dissertationsfrom the Faculty of Science and Technology 1390
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