Network Flow Model for the Simulation
of CO2 Sequestration Process
A Dissertation Presented
by
Daesang Kim
to
The Graduate School
in Partial Fulfillment of the
Requirements
for the Degree of
Doctor of Philosophy
in
Applied Mathematics and Statistics
Stony Brook University
August 2008
Stony Brook University
The Graduate School
Daesang Kim
We, the dissertation committee for the above candidate for the
Doctor of Philosophy degree,
Hereby recommend acceptance of this dissertation.
W. Brent Lindquist,
Dissertation Adviser, Professor, Applied Mathematics and Statistics, Stony
Brook University
Xiaolin Li,
Chairperson of Defense, Professor, Applied Mathematics and Statistics, Stony
Brook University
Joseph Mitchell,
Professor, Applied Mathematics and Statistics, Stony Brook University
Troy Rasbury
Associate Professor, Geosciences, Stony Brook University
This dissertation is accepted by the Graduate School
Lawrence Martin
Dean of the Graduate School
ii
Abstract of the Dissertation
Network Flow Model for the Simulation
of CO2 Sequestration Process
by
Daesang Kim
Doctor of Philosophy
in
Applied Mathematics and Statistics
Stony Brook University
2008
The pore-throat networks of porous rock samples are constructed from the 3D
analyses of the X-ray computed micro tomography (CMT) images of the samples and are
extended to network flow models in order to characterize the transport properties of flows
through porous media. Four major phases–void phase, kaolinite, quartz, and the minerals
of interest–are extracted by the backscattered electron (BSE) and energy dispersive X-ray
(EDX) analyses of three Viking samples; sandstone, shaly sandstone, and conglomerate
sandstone. The CMT images of the three Viking samples are segmented by repeated biphase segmentation to obtain the four-phase images, and the segmented images contain
the information about mineral abundances and accessibilities. The cluster sizes and
accessible surface areas of the minerals are computed, the size and area distributions of
iii
the minerals in the three Viking samples are obtained, and the characteristics of these
distributions for different rock samples are analyzed to estimate the reactivity of the rocks.
Furthermore, the pore-throat networks of mineral distributions constructed by similar 3D
image analyses are extended to network flow models for reactive flows using reaction
models and the identified minerals. Two types of the reaction models are used; kinetic
reactions and instantaneous reactions. The reaction rates of the kinetic reactions are
integrated over each time step, and the equilibrium of the instantaneous reactions is
satisfied at every time step. The network flow model is applied to the micro/pore scale
simulation of CO2 sequestration process in the Viking samples. This small scale
simulation reveals some pore-to-pore heterogeneities in the reaction rates, and the effects
of these heterogeneities are found to differ with the volume flow rate and the type of
Viking sample. The upscaled reaction rates are computed to understand the CO2
sequestration process and quantify the pore scale heterogeneities.
iv
Acknowledgement
v
Table of Contents
LIST OF FIGURES ................................................................................................................... VIII
LIST OF TABLES ........................................................................................................................ XI
I.
INTRODUCTION ................................................................................................................. 1
II.
NUMERICAL SCHEME .................................................................................................... 6
1.
3D image analysis.............................................................................................. 6
Segmentation......................................................................................................... 6
Medial axis ............................................................................................................. 7
Throat computation ............................................................................................... 8
Pore-throat network construction ......................................................................... 9
a.
b.
c.
d.
2.
Pore-throat network of mineral distributions .................................................. 10
a.
BSE and EDX analyses .......................................................................................... 10
b.
Mineral distributions of pore-throat network ..................................................... 11
3.
Single-phase network flow model ................................................................... 18
a.
Construction of the single-phase network flow model ....................................... 18
b.
Simplified model based on channel shape factor ............................................... 20
c.
Lattice Boltzmann Method .................................................................................. 23
4.
Network flow model for CO2 sequestration...................................................... 27
a.
The equation of the network flow model ........................................................... 27
b.
Kinetic reactions .................................................................................................. 29
c.
Equilibrium computation of instantaneous reactions ......................................... 31
d.
e.
f.
g.
III.
CO2 solubility in aqueous NaCl solution .............................................................. 40
Initial and boundary conditions ........................................................................... 42
Activity coefficients and diffusion coefficients.................................................... 43
Effective and volume-averaged reaction rates .................................................... 45
COMPUTATIONAL RESULTS ...................................................................................... 48
vi
1.
Pore-throat network of mineral distributions .................................................. 48
2.
Single-phase network flow model ................................................................... 65
3.
Simulation of CO2 sequestration ...................................................................... 68
a.
Validation of the reactive models ........................................................................ 68
b.
Application to Viking samples.............................................................................. 76
IV.
CONCLUSION & FUTURE STUDY .............................................................................. 94
vii
List of Figures
Figure 1. (a) The grey scale BSE image of Viking3W4. (b) The segmented image of the
grey scale BSE image. Black, green, grey, and red indicate void phase, minerals of MAN
< quartz, quartz, minerals of MAN > quartz. ............................................................ 11
Figure 2. One pore body contains kaolinite (green) inside and it neighbors two clusters of
minerals of interest (red). Black and grey colors represent void space and quartz space
respectively. S denotes surface areas between minerals or mineral and void space. V
denotes volumes of mineral clusters. ....................................................................... 15
Figure 3. (a) An example of constructed pore-throat network. (b) Network flow model is
developed from the pore-throat network. ................................................................. 18
Figure 4. Discrete velocity directions of LBM. N=19[27]. ......................................... 25
Figure 5. An example of the computation domain for the LBM. Throat, inlet, and outlet
voxels are shaded grey[27]. .................................................................................... 26
Figure 6. A schematic diagram of the simulation of CO2 sequestration. CO2 resolved
water is injected from the top boundary into the porous rock. A flow is induced
downward chemically interacting with minerals in the rock. ...................................... 27
Figure 7. One time step in the simulation ................................................................. 31
Figure 8. Histograms of the attenuation coefficients of three Viking samples with the
threshold windows. ............................................................................................... 50
Figure 9. (a) One slice of grey scale image of the conglomerate sandstone (Viking14W5).
(b) Bi-phase segmented image of the primary pore and primary grain spaces. Black and
white indicate the primary grain space and primary pore space respectively. (c)
Segmented image of kaolinite and void space in the primary pore space. Black and green
indicate the void space and kaolinite phase respectively. Grey indicates the primary grain
space. (d) Segmented image of quartz and MoI. Grey and red indicate quartz and MoI
respectively. White indicates the primary pore space. ................................................ 51
Figure 10. The final four-phase segmented image of the conglomerate sandstone
(Viking14W5) after adjustment step. Black, green, grey, and red indicate void, kaolinite,
quartz, and MoI respectively. .................................................................................. 52
Figure 11. Grey scale and four-phase segmented images of the sandstone (Viking3W4).
........................................................................................................................... 52
Figure 12. Grey scale and four-phase segmented images of the shaly sandstone
(Viking10W4). ...................................................................................................... 53
viii
Figure 13. Statistics of the pore-throats networks of the primary pore space of the three
Viking samples. The sandstone data are represented by solid lines and ‘•’. The shaly
sandstone data are represented by dotted lines and ‘x’. The conglomerate sandstone data
are represented by dashed lines and ‘□’. ................................................................. 55
Figure 14. Statistics of the mineral abundances and accessibilities of the sandstone,
Viking3W4. .......................................................................................................... 59
Figure 15. Statistics of the mineral abundances and accessibilities of the shaly sandstone,
Viking10w4. ......................................................................................................... 60
Figure 16. Statistics of the mineral abundances and accessibilities of the conglomerate
sandstone, Viking14W5. ........................................................................................ 61
Figure 17. Percent pore volumes and percent pore surface areas of kaolinite and MoI.
Sandstone, Viking3W4. ......................................................................................... 62
Figure 18. Percent pore volumes and percent pore surface areas of kaolinite and MoI.
Shaly sandstone, Viking10w4. ................................................................................ 63
Figure 19. Percent pore volumes and percent pore surface areas of kaolinite and MoI.
Conglomerate sandstone, Viking14w5. .................................................................... 64
Figure 20. Pressure distribution. FB22. Ptop=101.3 and Pbot=0. ................................... 66
Figure 21. Pressure distribution. Ptop=101.3 and Pbot=0. ............................................. 67
Figure 22. Permeabilities with respect to weight factor, ω[15]. ................................... 67
Figure 23. Graphs of the total mass change rates and effective reaction rates. .............. 74
Figure 24. Histograms of pH, [Ca2+], SIA, SIK, and reaction rates at the steady state. .... 75
Figure 25. pH plots in the primary pore space of Viking3W4. Grey and dark grey
represent quartz and MoI respectively. The CO2 resolved saline is injected from top to
bottom with 𝒗𝒔𝒆𝒆𝒑 = 𝟎. 𝟎𝟎𝟓𝟖𝐜𝐦/𝐬. .................................................................... 81
Figure 26. pH plots in the primary pore space of Viking10W4. Grey and dark grey
represent quartz and MoI respectively. The CO2 resolved saline is injected from top to
bottom with 𝒗𝒔𝒆𝒆𝒑 = 𝟎. 𝟎𝟎𝟓𝟖𝐜𝐦/𝐬. .................................................................... 82
Figure 27. pH plots in the primary pore space of Viking14W5. Grey and dark grey
represent quartz and MoI respectively. The CO2 resolved saline is injected from top to
bottom with 𝒗𝒔𝒆𝒆𝒑 = 𝟎. 𝟎𝟎𝟓𝟖𝐜𝐦/𝐬. .................................................................... 83
Figure 28. Mass change rates and effective reaction rates of anorthite and kaolinite.
Viking3W4. 𝒗𝒔𝒆𝒆𝒑 = 𝟎. 𝟎𝟎𝟓𝟖𝐜𝐦/𝐬. The dotted lines in the graphs of the effective
reaction rates indicate the volume-averaged reaction rates. ........................................ 84
Figure 29. Mass change rates and effective reaction rates of anorthite and kaolinite.
Viking3W4. 𝒗𝒔𝒆𝒆𝒑 = 𝟎. 𝟎𝟎𝟏𝐜𝐦/𝐬. The dotted lines in the graphs of the effective
ix
reaction rates indicate the volume-averaged reaction rates. ........................................ 85
Figure 30. Mass change rates and effective reaction rates of anorthite and kaolinite.
Viking3W4. 𝒗𝒔𝒆𝒆𝒑 = 𝟎. 𝟎𝟎𝟎𝟏𝐜𝐦/𝐬. The dotted lines in the graphs of the effective
reaction rates indicate the volume-averaged reaction rates. ........................................ 86
Figure 31. Histograms of pH, [Ca2+], SI’s, and reaction rates at t=810sec. Viking3W4.
𝒗𝒔𝒆𝒆𝒑 = 𝟎. 𝟎𝟎𝟓𝟖𝐜𝐦/𝐬. ....................................................................................... 87
Figure 32. Histograms of pH, [Ca2+], SI’s, and reaction rates at t=1223sec. Viking3W4.
𝒗𝒔𝒆𝒆𝒑 = 𝟎. 𝟎𝟎𝟏𝐜𝐦/𝐬. ......................................................................................... 88
Figure 33. Histograms of pH, [Ca2+], SI’s, and reaction rates at t=1223sec. Viking3W4.
𝒗𝒔𝒆𝒆𝒑 = 𝟎. 𝟎𝟎𝟎𝟏𝐜𝐦/𝐬. ....................................................................................... 89
Figure 34. Mass change rates and effective reaction rates of anorthite and kaolinite.
Viking14W5. 𝒗𝒔𝒆𝒆𝒑 = 𝟎. 𝟎𝟎𝟓𝟖𝐜𝐦/𝐬. The dotted lines in the graphs of the effective
reaction rates indicate the volume-averaged reaction rates. ........................................ 90
Figure 35. Mass change rates and effective reaction rates of anorthite and kaolinite.
Viking14W5. 𝒗𝒔𝒆𝒆𝒑 = 𝟎. 𝟎𝟎𝟏𝐜𝐦/𝐬. The dotted lines in the graphs of the effective
reaction rates indicate the volume-averaged reaction rates. ........................................ 91
Figure 36. Histograms of pH, [Ca2+], SI’s, and reaction rates at t=1223sec. Viking14W5.
𝒗𝒔𝒆𝒆𝒑 = 𝟎. 𝟎𝟎𝟓𝟖𝐜𝐦/𝐬. ....................................................................................... 92
Figure 37. Histograms of pH, [Ca2+], SI’s, and reaction rates at t=1223sec. Viking14W5.
𝒗𝒔𝒆𝒆𝒑 = 𝟎. 𝟎𝟎𝟏𝐜𝐦/𝐬. ......................................................................................... 93
x
List of Tables
Table 1. Equilibrium and reaction rate constants of anorthite and kaolinite reactions .... 30
Table 2. Instantaneous reactions and equilibrium constants of the reactions ................. 32
Table 3. 9 derived instantaneous reaction, and corresponding equilibrium equations to
compute the activities of the 9 non-component species. ............................................. 33
Table 4. Total concentrations of all components. ....................................................... 34
Table 5. Table for the equilibrium computation. ........................................................ 36
Table 6. Table of temperature vs. dielectric constant of pure water. ............................. 44
Table 7. Threshholds, T0 and T1 for the bi-phase segmentations.................................. 49
Table 8. Mineral abundances and accessibilities of three Viking samples. The abundances
and accessibilities are compared with those by BSE analysis. .................................... 53
Table 9. the numbers of pores which contain kaolinite, or MoI. .................................. 57
Table 10. Mean and standard deviations of pores, kaolinite clusters, and MoI clusters. . 58
Table 11. Computation result of permeabilities, KLB, KSF, and KBZ[15]. ...................... 66
Table 12. Summary of the generated network. .......................................................... 70
Table 13. Three reactions for the condition of the top boundary inflow. ....................... 71
Table 14. Initial state of pores. pH is 6.6. ................................................................. 71
Table 15. The condition of the top boundary inflow. The CO2 solubility is 2.0M and pH is
2.9. ...................................................................................................................... 72
Table 16. Effective reaction rates, 𝑹𝑵, volume-averaged reaction rates, 𝑹, and the ratio
𝜷 = 𝑹/𝑹𝑵. The result is compared with Li. ............................................................ 73
Table 17. The condition of the top boundary inflow. CO2 solubility is 1.01M and pH is
3.01. .................................................................................................................... 77
Table 18. Simulation result of the effective and volume-averaged reaction rates. The
reaction rates are computed at the times given in the table. ........................................ 80
xi
I.
Introduction
The use of fossil carbon resources has been generating large amount of CO2,
which is the main cause of the global warming. There are many intense researches to
reduce this massive CO2 released to the atmosphere by storing the CO2 in subsurface [1;
2; 3; 4]. The CO2 sequestration consists of the basic four processes; capture, transport,
storage, and monitoring. The capture is the process of extracting CO2 from the
hydrocarbon combustion and the carbon storage is the process of injecting the captured
CO2 in storage locations such as deep saline aquifers, depleted oil/gas reservoirs,
unminable coal seams, etc. The deep saline aquifers are rich and widely distributed, and
thus, massive amount of CO2 can be stored in the aquifers. Furthermore, the choice of
deep saline aquifers is economically efficient because there are more chances of
appropriate groundwater aquifers near the CO2 sources such as industrial power plants to
minimize the transport costs. To store large amount of CO2 safely and efficiently in the
aquifers for long times, the subsurface reactive flows in saline aquifers must be
completely understood. The reaction kinetics of the CO2 sequestration process is affected
by the subsurface flow through porous rocks and the effect of pore-pore heterogeneities
emerges in the reaction kinetics. It is known that many lab experiments in which mineral
samples are usually crushed and mixed exclude cementation effect and porous structure
of sedimentary rocks, and this may cause large difference between reaction rates from lab
experiments and field experiments [5]. The reactive flow analysis should be performed at
the pore scale to catch the micro scale effects, and the upscaling the pore scale analysis to
obtain parameters for continuum scale simulation follows [4] [6].
1
There are a few researches of pore scale simulation of reactive subsurface flows.
Pore scale flows of CO2 sequestration are simulated by using network flow modeling, and
a methodology is suggested for the pore scale simulation and upscaling from the micro
scale results to macro scale parameters [4]. The network flow model is one of the most
practical tools to analyze fluid flows through porous media, and it can handle singlephase flows, multi-phase flows, and multi-species reactive flows. It requires a simplified
pore-channel network which is extracted from the pore space and stores the fluid
transport property of the porous rock, and it cannot include the detail geometry of the
porous rock. Thus, it can handle somewhat large size of a rock sample: usually, a rock
sample of a few-milimeter size. For the simulation of reactive flows, reactive mineral
information is also required. The network used for the simulation is not a network
constructed directly from a real rock, but a regular lattice network artificially generated
based on statistical models. The reaction rate models of two minerals, anorthite and
kaolinite are used, and nine chemical reactions are taken into consideration for the
simulation. The simulation shows that the spatial heterogeneities of the reaction rates
exist and that the network flow model is a proper method to capture the heterogeneities in
the CO2 sequestration process. However, the properties of the reaction heterogeneities
highly depend on the network data and the reaction model. The approach is based on a
plausible reaction model, but it is still unknown how realistic the artificially generated
network is.
Another research uses the lattice Boltzmann method to simulate pore scale
reactive flows [6; 7]. The LB method is very advantageous for flows through porous
media. It is simple and flexible, and so, it is appropriate for an irregular domain like
porous media. Segmented images can be directly computation domain without surface
2
construction and grid generation which are time consuming steps in the computational
fluid dynamics. It is almost the unique direct simulation method for fluid flows through a
porous media of core size. The LB method is available for both single-phase and twophase flow analyses to compute the absolute permeability and relative permeability of a
porous rock [8; 9]. The LB method does not simplify the pore space like the network flow
model, and so, the result of the LB method contains whole information of pore geometry.
Moreover, it does not need the analyses of pore space such as medial axis computation,
throat computation, and pore-throat network construction which are necessary for the
network flow model. The LB method captures fluid flow more directly and accurately,
and the most advantage of the method is that it can simulate dissolution and precipitation.
The LB method is applied for a simple single-phase flow through a porous rock of core
size, but for a more complex reactive flow such as CO2 sequestration, the method is
limited by the huge computation burden of the LB simulation. So far, the LB method is
applied for pore scale reactive flows through 2-dimensional porous media, or simple 3dimensional domains.
The synchrotron technology to generate X-ray computed tomography of rock
samples and 3DMA-ROCK, a software package for 3D image analyses of the CMT
images enable a direct geometrical analysis of porous structure of sedimentary rocks [10;
11; 12; 13; 14]. 3DMA-ROCK constructs the pore-throat network of the sample rocks by
segmentation, medial-axis computation, throat finding, and pore space partitioning, and
the pore-throat network is basis for the network flow models. The constructed pore-throat
network of a real sample rock can be extended to the network flow model for singlephase flow using a simple channel conductance model or the Lattice Boltzmann method
for channel conductances. The single-phase network flow model is available to compute
3
the absolute permeability, volume flow rates though all channels, and pressure
distribution of all pores, which are required for a simple transport flow analysis[15; 16].
3DMA-ROCK is used to identify and analyze images of more than 2 phases: rock, water,
and oil [17], and it can be used for a multi-phase image which stores not only porous
structure but also mineral distributions of a rock sample. The backscatter electron (BSE)
and energy dispersive X-ray (EDX) analyses are used to identify chemical properties and
compositions of each mineral phase which is identified by 3DMA-ROCK [18].
The goal of this research is to develop a methodology to simulate CO2
sequestration in a real porous rock and to estimate the upscaled parameters for the CO2
sequestration process in the real porous rock. We should perform a pore scale simulation
in a core size domain to estimate the upscaled parameters, and thus, we select the
network flow model for the reactive flow simulation. Therefore, we extend the 3DMA
ability to explore not only the porous structure but the mineral identification and
composition of a real rock sample, and furthermore, we construct a pore-throat network
of mineral distributions. The four major phases – void, kaolinite, quartz, and minerals of
interest – are determined for mineral identification of a sample rock by the BSE and EDX
analyses, and four phase images segmented by repeated bi-phase segmentations of the
CMT images are obtained. We can identify the mineral compositions and the accessible
mineral clusters with the four-phase segmented images, and we can compute the mineral
surface areas accessible from void space to quantify the accessibility of each mineral
phase and to characterize the mineral distributions. A pore-throat network of mineral
distributions is constructed, and the network is used to develop the network flow
simulation for CO2 sequestration. We can simulate the sequestration process with the
constructed network of the real rock eliminating one uncertainty of the artificial network
4
which might be unrealistic, and estimate more realistic upscaled parameters from the pore
scale simulation. The network simulation is applied to different types of rocks, and the
effect of different rocks is also obtained.
5
II.
Numerical scheme
1. 3D image analysis
a. Segmentation
Segmentation is an image processing to obtain a binary (or more phase) image
from a grey-scale image. The kriging based algorithm is used to segment the grey-scale
CMT images[14], and the method requires two thresholds values, T0 and T1. Voxels of
intensity less than T0 are assigned to one phase, Π0 of the two phases and voxels of
intensity greater than T1 are assigned to the other phase, Π1. The other voxels of intensity
between T0 and T1 are determined by the kriging algorithm, which uses indicator
variables, 𝑖(𝑧𝑐 ; 𝑥𝛼 ).
𝑖(𝑧𝑐 ; 𝑥𝛼 ) = {
if 𝑧(𝑥𝛼 ) ≤ 𝑧𝑐
otherwise
1,
0,
(1)
𝑧(𝑥𝛼 ) is the attenuation coefficient at the position, 𝑥𝛼 , and 𝑧𝑐 is a certain threshold
value. The probability of 𝑧(𝑥0 ) ≤ 𝑧𝑐 for an unidentified voxel, 𝑥0 is estimated by the
unbiased linear estimator of the indicator variable, 𝑖(𝑧𝑐 ; 𝑥0 ).
𝑛
𝑃(𝑧(𝑥0 ) ≤ 𝑧𝑐 ) = ∑ 𝜆𝛼 (𝑧𝑐 ; 𝑥0 )𝑖(𝑧𝑐 ; 𝑥𝛼 )
(2)
𝛼=1
The covariance of the indicator variables at two different points is denoted by 𝐶(𝑧𝑐 ; 𝑥𝛼 −
6
𝑥𝛽 ) = Cov(𝑖(𝑧𝑐 ; 𝑥𝛼 ), 𝑖(𝑧𝑐 ; 𝑥𝛽 )) . The above 𝜆𝛼 (𝑧𝑐 ; 𝑥0 ) are obtained by solving the
below kriging system.
𝑛
∑ 𝜆𝛽 (𝑧𝑐 ; 𝑥0 )𝐶(𝑧𝑐 ; 𝑥𝛼 − 𝑥𝛽 ) + 𝜇(𝑧𝑐 ; 𝑥0 ) = 𝐶(𝑧𝑐 ; 𝑥𝛼 − 𝑥0 ) for 𝛼 = 1, ⋯ , 𝑛
𝛽=1
(3)
𝑛
∑ 𝜆𝛽 (𝑧𝑐 ; 𝑥0 ) = 1
𝛽=1
𝛼(= 1, ⋯ , 𝑛) represents neighborhood voxels of 𝑥0 . For each unidentified voxel, 𝑥0 ,
the two probabilities, 𝑃1 = 𝑃(𝑧(𝑥0 ) ≤ 𝑇0 ) and 𝑃2 = 𝑃(𝑧(𝑥0 ) ≥ 𝑇1 ) = 1 − 𝑃(𝑧(𝑥0 ) ≤
𝑇1 ) are estimated by the above equation (2). If 𝑃1 ≥ 𝑃2 , the voxel, 𝑥0 is identified as
the phase Π0. Otherwise, the voxel, 𝑥0 is identified as the phase Π1.
After the segmentation, isolated grain voxels are cleaned up because the isolated
grain voxels are physically unrealistic. Isolated pore voxels are also cleaned because the
isolated pore space is not accessible. For example, any fluid flow cannot reach the
isolated pore space and the space is cleaned to make the further analyses simple.
b. Medial axis
The media axis is a percolation backbone of void space and it is a lower
dimensional structure containing the basic topological characteristics of the porous
media[10]. The medial axis has the same topological properties as the whole porous
media and so, it is useful for the further analyses because of the simple and lower
dimensional structure. A medial axis is obtained by thinning the given void space until
7
the space becomes a set of curves and the Lee-Kashyap-Chu algorithm is used for the
medial axis[19].
The basic medial axis computation of a digitized image is very sensitive to the
surface noise and there are a lot of unwanted medial axis paths obtained and a trimming
step of the useless paths is needed. All branch-leaf paths of length less than the maximum
burn number of the path voxels are deleted by the trimming step. Besides, other unwanted
paths such as leaf-leaf paths and needle-eye paths are deleted[10].
Each branch clusters will become a pore body after throat computation and porethroat network construction, but some branch clusters are not appropriate for a pore body
if they are too close to each other. Some of the close clusters should be merged into one
branch cluster. If the distance between two branch clusters is less than the burn distances
of the two branch clusters, then the two branch clusters are merged into one branch
cluster.
c. Throat computation
For each medial axis path, a throat of the path is the cross section of minimum
area along the medial axis path and the throat is important for a fluid flow analysis
because it is a bottleneck of the flow along the medial axis. Thus, the throats are also used
to partition the whole void space into pore bodies. There are a few algorithms for the
throat computation, the wedge based algorithm and the Dijkstra shortest path based
algorithm[10; 13]. The wedge based algorithm constructs a local grain boundary for each
medial axis path by the grass fire and searches a rough perimeter on the local grain
boundary for each medial axis voxel. The algorithm compares cross section areas of the
8
rough perimeters to find the minimum cross section area. The perimeter of minimum
cross section area is the basis of the throats of the medial axis and the throat perimeter is
constructed by connecting the voxels of the rough perimeter. The Dijkstra based
algorithm dilates the medial axis path by one voxel distance in the L∞ direction and
searches any perimeter of grain voxels touching the swelling medial axis path voxels. If it
finds an enclosing perimeter, the algorithm finds the shortest perimeter among the
enclosing perimeters and the shortest perimeter is the throat perimeter of the medial axis
path. The two methods are used to compute throats and two sets of throats are merged
into a full set of throats.
d. Pore-throat network construction
To construct a pore-throat network, the void space is partitioned into pore bodies
by grain boundaries and throat barriers. Each isolated void space becomes a pore body
and pore bodies can neighbor each other through a throat. The pore partitioning algorithm
allows each throat touch only two pore bodies. If a throat touches only one pore body or
more than two pore bodies, the throat is deleted. If a throat touches more than two throats,
then the throat crosses another throat. Thus, a throat of a crossed pair is deleted until all
throats touch exactly two throats. The current algorithm allows the loss of some throats to
construct a reasonable pore-throat network.
9
2. Pore-throat network of mineral distributions
a. BSE and EDX analyses
The scanning electron microscope (SEM) is a type of microscopes using electrons
to make the images of a specimen surface. The SEM is used in backscatter mode to
produce the backscatter electron (BSE) images. The SEM generates the BSE images by
measuring the amount of electrons backscattered which is proportional to the mean
atomic number (MAN). The core sample can be imaged by the SEM to identify the
mineral composition on the sample surface by measuring the MAN.
The energy dispersive X-ray (EDX) analysis generates an energy spectrum of
emitted X-ray for a point on the core surface. The EDX is used to identify the mineral on
each point and moreover, the chemical composition of the mineral. We construct a 2dimensional image of minerals over a certain region on the core surface by combining
BSE and EDX analyses.
The below images are sample BSE images of Viking 3W4 and the most dominant
mineral of the rock is quartz. The minerals of MAN less than MAN of quartz and MAN
greater than that of quartz are identified, and the segmented BSE image of the Viking
3W4 sample is shown in figure. Black, green, grey, and red represent void space,
minerals of MAN less than quartz, quartz, and minerals of MAN greater than quartz. By
the EDX analysis, the minerals in each color are identified. Green color includes kaolinite,
albite, Mg chlorite, etc, and red color includes anorthite, K-feldspar, apatite, etc. In the
green phase, the kaolinite is most prominent and so, the green phase is named kaolinite.
The red is named the minerals of interest because the phase has some minerals reactive
with the injected saline[18].
10
(a)
(b)
Figure 1. (a) The grey scale BSE image of Viking3W4. (b) The segmented
image of the grey scale BSE image. Black, green, grey, and red indicate
void phase, minerals of MAN < quartz, quartz, minerals of MAN > quartz.
b. Mineral distributions of pore-throat network
In the previous section a. BSE and EDX analyses, there are four major phases,
void space, kaolinite, quartz, and the minerals of interest, and the CMT images are
segmented to identify the four phases. Three steps of bi-phase segmentations are used to
11
assign one of four phases to each voxel and the kriging based algorithm is also used for
each step of bi-phase segmentation[14]. First, we apply the bi-phase segmentation to
identify the primary pore space (void space + kaolinite) and the primary grain space
(quartz + the minerals of interest). Second, we use the algorithm again to indentify the
void space and kaolinite in the obtained primary pore space ignoring the information
outside the primary pore space. This method avoids losing void phase, the least X-ray
attenuated phase by blocking the effect of higher attenuated phases-kaolinite and the
minerals of interest- around the void phase. Likewise, the bi-phase segmentation is
applied to identify kaolinite and the minerals of interest in the segmented primary grain
space.
The resolution, 3.98μm of the CT images is larger than the resolution, 1.8μm of
the BSE images for the mineralogical analysis, and thus, we have more possibility to lose
some voxels of the most attenuated phase, the minerals of interest and the least attenuated
phase, void phase because the value of the intensity of one voxel is averaged over the
volume of the voxel. This causes smaller amounts of the void space and the minerals of
interest, and especially, smaller surface areas between the two phases than those from the
BSE analysis. We lose many voxels of the void phase and the minerals of interest, and
this occurs often when the void phase and the minerals of interest are adjacent to each
other. We lose not only the voxels of the void phase and the minerals of interest but also
the surface area between the two phases which is most important in the mineral
accessibility analysis. An adjustment step of the two phases is needed to obtain more
consistent segmented images with the BSE result. The space of the minerals of interest is
dilated outward by one- or two-voxel distance and a burned voxel is flipped to the phase
of the minerals of interest if the burned voxel neighbors a void voxel. If it does not
12
neighbor any void voxel, the burned voxel remains to be of the original phase. This
process can remove some voxels of incorrectly assigned kaolinite and quartz phases, and
it can recover voxels of the void phase and the minerals of interest and the inter-phase
surface area between void phase and the minerals of interest. We apply a similar
procedure to the void phase to increase the void space near the minerals of interest in
one- or two-voxel distance.
Next, the segmented images are cleaned up by removing some unrealistic or
inappropriate voxels. Any mineral voxel inside the void space is physically unrealistic
and the floating voxel is cleaned up to the void phase. All isolated primary pore voxels
(void space + kaolinite) surrounded by the primary grain space (quartz + the minerals of
interest) of water-blocking materials are cleaned up to the quartz phase because a fluid
flow cannot reach the isolated primary pore regions. For the further analysis, we use two
types of segmented images, four-phase segmented images for mineralogical analysis and
bi-phase segmented images for pore-throat network construction. The above cleaned
images are the final segmented images for the further mineralogical analysis, but the biphase images of the primary pore phase and primary grain phase are cleaned up again
deleting all primary grain voxels floating in the primary pore space. Thus, the final
images for mineral information and those for pore-throat network are slightly different
and the difference is very small, e.g., it is less than 0.1% in the case of Viking3W4.
However, we use the both images when we develop a pore-throat network of mineral
distributions, and so, all mineral information can be stored in the constructed pore-throat
network of mineral distributions.
After the segmentation, the pore-throat network of the primary pore space is
constructed to explore the overall mineral distribution of the Viking samples on the basis
13
of each pore in the pore-throat network. As shown in the previous sections, there are a
few steps to construct the pore-throat network; medial axis construction, medial axis
trimming, throat computation, and partitioning the pore space.
For each pore body of the constructed pore-throat network of the primary pore
space, we search the mineral distributions of the pore by computing the amount of
minerals inside the pore body and touching the pore body and by computing the surface
areas between minerals in the pore and between the pore body and minerals. We combine
the mineral distributions of each pore body with the pore-throat network of the primary
pore space to construct a pore-throat network of mineral distributions for further analyses,
such as network flow simulation of the CO2 sequestration process in the porous rock. The
Figure 2 shows a simplified geometry around a pore body of the constructed pore-throat
network of mineral distributions. Four colors, black, green, grey, and red represent four
phases, void, kaolinite, quartz, and the minerals of interest respectively. The largest circle
in the middle of the figure is a pore body under consideration, and the pore body has
some void phase and some kaolinite phase. It is connected to three other pores by three
throat channels and it touches two red clusters. Most part of the pore is surrounded by the
dominant phase, quartz. First, various volumes are computed: the pore volume, kaolinite
volume, VG in the pore, and volumes, VR1, VR2 of the minerals of interest are computed.
Next, the surface areas accessible from the pore body are computed: the area, SRB of the
minerals of interest exposed to void and the area, SGB of the kaolinite exposed to void are
computed. SRB and SGB are mineral surface areas accessible directly from the pore. The
area of the minerals of interest coated by the kaolinite is also computed and this surface
area is also accessible through the kaolinite phase. Some other surface areas such as the
14
area, SQB between void phase and quartz and the area, S GQ between kaolinite and quartz
are also obtained.
Figure 2. One pore body contains kaolinite (green) inside and it neighbors two clusters of
minerals of interest (red). Black and grey colors represent void space and quartz space
respectively. S denotes surface areas between minerals or mineral and void space. V denotes
volumes of mineral clusters.
The volumes of pore body, void phase, kaolinite, and the minerals of interest are
simply computed by counting the number of voxels of the corresponding phases and the
surface areas is computed by the marching cube. The marching cube is a frequently used
algorithm for an inter-phase surface construction in a bi-phase image [20; 21]. The
algorithm uses a dual grid and it constructs and triangulates surfaces by inspecting all
eight edges of each cube of the dual grid. There are 256(=28) total possible configurations
but there are only 15 unique configurations. The other configurations other than the 15
configurations are generated by rotation and reflection of the 15 configurations. The areas
15
of the inter-phase surface can be computed based on the constructed surface triangles. For
the four-phase images, a few steps of the marching cube are needed to construct the interphase surface between two phases. Let P1 and P2 be the two phases of which inter-phase
surface in the four-phase image is to be computed. Let B1 and B2 be the two phases of a
bi-phase image which is to be artificially constructed for the computation of the interphase surface area. First, a new bi-phase image is constructed by assigning B1 to the
voxels of P1 and B2 to the all voxels other than P1. The marching cube algorithm is
applied to the bi-phase image for construction of the inter-phase surface of B1 and B2. Let
S1 be the constructed inter-phase surface. Likewise, the other bi-phase image is
constructed by assigning B2 to the voxels of P2 and B1 to the other voxels and the
marching cube is applied for the inter-phase surface, S2 between B1 and B2. The two
surfaces, S1 and S2 are compared to find common vertices and the inter-phase surface of
P1 and P2 is defined by the common vertices. The surface area is obtained by the triangles
of the common vertices. The computation loses some boundary areas while the interphase surface is constructed with the common vertices and this causes some error in the
surface area.
The surface area of a digitized object can vary widely from the real surface area
with the resolution of image while the volume of a digitized object converges stably to
the true volume. The result of the above area cannot be used directly and it is corrected
by the BET surface area. The BET (Brunauer, Emmett and Teller) method is based on the
multilayer adsorption of gas molecules on a solid surface[22]. It estimates the number of
adsorbed molecules in the first layer on the solid surface and computes the surface area
by the number of molecules of the first layer. The BET surface area of the Viking 3W4
sample is measures and the area to volume ratio is approximately 40,000cm-1. Total
16
surface area of the pore space is corrected by multiplying a constant factor to make the
surface to volume ratio, S/V= 40,000cm-1. The surface areas of the other rock samples are
also corrected by multiplying the factor.
17
3. Single-phase network flow model
a. Construction of the single-phase network flow model
Based on the pore-throat network computed by the previous procedure, we can
easily construct the single-phase network flow model[15; 16]. Most of network flow
models simplify the pore space to a network of pores and channels. Only a pore body
contains the fluid, and the fluid can flow only through a channel. A channel connects two
pores allowing the fluid to flow from one of the pores to the other pore, and the channel
volume is ignored. A pore of the network flow model is a pore from the pore-throat
network and a channel of the network flow model is pore space around the corresponding
throat. Thus, the properties of a pore in the network flow model are determined by the
corresponding pore of the pore-throat network, and those of a channel are determined
mainly by the corresponding throat and partially by the two pores which the throat
connects.
(a)
(b)
Figure 3. (a) An example of constructed pore-throat network. (b) Network flow model is
developed from the pore-throat network.
18
The flow through the whole network is driven by the pressure difference between
one side of boundaries and the opposite side. We assume a pressure of a pore is constant
in the pore and the flow through a channel is caused by the pressure difference between
the pores which the channel connects. The goal of the single-phase network flow model is
to compute the absolute permeability of the rock, pressure of pores, and flow rates
through all channels. The absolute permeability of a rock is defined by the below
equation[23].
𝐾=
𝜇𝑄𝐿
𝐴𝛥𝑃
(4)
μ is the viscosity of the fluid, Q is the volume flow rate through the porous rock, A is the
cross-section area of the sample normal to flow direction, L is the length of the rock in
the flow direction, and ΔP is the pressure difference between the inlet and outlet
boundaries. From a simple channel flow theory, the flow rate through a channel is
proportional to the pressure difference between the ends of the channel, and the channel
conductance from pore i to pore j is defined by
𝑄𝑖𝑗 = 𝐶𝑖𝑗 (𝑃𝑖 − 𝑃𝑗 )
(5)
Qij is the volume flow rate from pore i to pore j, and Pi and Pj are pressures of pore i and
pore j respectively. For each pore i, the net flow rate should be zero to satisfy mass
conservation.
19
∑ 𝑄𝑖𝑗 = 𝐶𝑖𝑗 (𝑃𝑖 − 𝑃𝑗 ) = 0
(6)
𝑗
j is one of pore indices of neighboring pores. Let the fluid injected from top boundary to
bottom boundary, and let Ptop and Pbot be pressures on top and bottom boundaries
respectively. By the boundary condition, the pore pressures of the top boundary are set to
Ptop and the pore pressure of the bottom boundary are set to Pbot. The pressures of the
pores that do not touch the top boundary or the bottom boundary are unknown. Let Np be
the number of the “vertically interior” pores that do not touch the top or the bottom
boundaries. We apply the equation for the all vertically interior pores and we have a
system of Np linear equations with Np unknown pore pressures. The channel which
connects pore i to pore j also connects pore j to pore i, and thus, Cij=Cji. The linear system
is symmetric, and moreover, the matrix is a real, sparse, diagonally dominant matrix. The
sparse, preconditioned conjugate-gradient is applied to solve the linear system to obtain
the pore pressure[24] [25]. With the computed pore pressure, the volume flow rate
through each channel is computed by the equation (5), and the whole volume flow rate
and the absolute permeability are also computed by the equation (4). However, there are
two methods to compute the channel conductance, Cij.
b. Simplified model based on channel shape factor
We assume the cross section of a channel is constant. Then, we can obtain the
channel conductance analytically by solving the 2-dimensional elliptic Poisson equation
which implied the Poiseuille’s flow through the channel[26].
20
𝜇∇2 𝑢
⃗ − ∇𝑃 = 0,
(7)
𝑢
⃗ = 0 on the wall
𝑢
⃗ is the distribution of the flow velocity on the channel cross section and ∇𝑃 is the
pressure gradient in the flow direction. ∇𝑃 is constant because the channel flow is
assumed to be fully developed. The Poisson equation is solved numerically for the
conductance of various triangular cross sections and a simple equation based on the many
numerical solutions is proposed for the conductances of triangular channels[16].
𝑔̃ = 𝑔
𝜇
3
≈ 𝐺
2
𝐴
5
(8)
𝑔̃ is the dimensionless hydraulic conductance, 𝐴 is the cross section area, and G is the
shape factor, G=A/P2. P is the perimeter of the channel cross section. Similarly, we derive
a cross sectional shape factor of a pore, 𝐺𝑝 =
𝑉𝐷
𝑆2
where 𝑉 is the pore volume, 𝐷 is a
3
pore diameter, and 𝑆 is the pore surface area. We use the equation, 𝑔̃𝑝 ≈ 5 𝐺𝑝 for the
dimensionless hydraulic conductance of a pore with the assumption of triangular cross
section. The maximum shape factor of triangles is the shape factor of the equilateral
triangle, 𝐺𝑇,𝑚𝑎𝑥 = √3/36. For a throat of the shape factor greater than 𝐺𝑇,𝑚𝑎𝑥 , the
above the equation of triangular cross section is not appropriate, and two new relations of
the shape factor greater than 𝐺𝑇,𝑚𝑎𝑥 are made based on the analytical solutions of
rectangular and elliptic cross sections. The below simple linear equations are suggested
for different values of the shape factor to compute the dimensionless hydraulic
conductance[15].
21
0.6𝐺,
𝑔̃ = { 0.5623𝐺,
0.5𝐺,
0 ≤ 𝐺 ≤ 𝐺𝑇,𝑚𝑎𝑥
𝐺𝑇,𝑚𝑎𝑥 ≤ 𝐺 ≤ 𝐺𝑅,𝑚𝑎𝑥
𝐺𝑅,𝑚𝑎𝑥 ≤ 𝐺 ≤ 𝐺𝐸,𝑚𝑎𝑥
(9)
𝐺𝑇,𝑚𝑎𝑥 , 𝐺𝑅,𝑚𝑎𝑥 , and 𝐺𝐸,𝑚𝑎𝑥 are the maximum shape factors of triangles, rectangles, and
ellipses. 𝐺𝑇,𝑚𝑎𝑥 = √3/36 when the triangle is equilateral. 𝐺𝑅,𝑚𝑎𝑥 =1/16 when the
rectangle is equilateral. 𝐺𝑇,𝑚𝑎𝑥 = 1/4𝜋 when the ellipse is circular. The above equation
is used for the dimensionless hydraulic conductances of both throat and pore.
The channel conductance, Cij is expressed by the hydraulic conductance of the
throat and two connected pores.
𝑑𝑡 1 𝑑𝑖 𝑑𝑗
𝐶𝑖𝑗 = ( + ( + ))
𝑔𝑡 2 𝑔𝑖 𝑔𝑗
−1
(10)
dt, di, and dj are the distances of the throat, pore i, and pore j respectively. The total
distance of the throat and the two pores is the length, d0 of the corresponding medial axis
path. We use a weight, ω to assign the throat distance, and we set the pore distances
proportional to the pore diameters. Then, the 3 distances are
𝑑𝑡 = 𝜔𝑑0 ,
𝑑𝑖 =
(1 − 𝜔)𝐷𝑖
,
𝐷𝑖 + 𝐷𝑗
𝑑𝑗 =
(1 − 𝜔)𝐷𝑗
𝐷𝑖 + 𝐷𝑗
A proper choice of ω is required for the single-phase network and some experimental
data are used to estimate the best value of ω.
22
c. Lattice Boltzmann Method
In the previous section, we use a strong assumption: the channel is a triangular,
rectangular, or elliptic cylinder. The assumption makes the computation simple and
efficient, but the simplification may cause some unknown error. For a more accurate
channel conductance, we must consider the real geometry of the throat and two connected
pores. Another approach for this purpose is the Lattice Boltzmann Method (LBM) [8; 15;
27; 28] [29]. The LBM is a fluid simulation technique like classical finite difference
(FDM), finite volume (FVM), and the finite element (FEM) methods. The FDM and
FEM solve a system of partial differential equations, the Navier-Stokes’ equation, or
Stokes’ equation which is a simplified version of the Navier-Stokes’s equation for a
creeping flow such as ground water/oil flows. The FVM solves a system of integral
equations of the Navier-Stokes’ equation. While the Navier-Stokes’ equation is based on
the macroscopic description of fluid continuum, the LBM is partially of microscopic
properties. The LBM is not fully microscopic, i.e., it does not contain whole microscopic
molecular dynamics. Thus, the LBM is called mesoscopic. The LBM has a particle
distribution on the lattice and the particle distribution involves some discrete velocity
directions up to 27. During each time step, the particle function is updated obeying a
scatter rule. There are a few LBM models including the most popular LBM model, the
lattice Boltzmann Bhatagnar-Gross-Krook (LBGK). The LBGK model with a curved
boundary condition is applied to compute the channel conductance in the second-order
accuracy.
23
𝑓𝑖 (𝑥 + 𝑒𝑖 Δ𝑡, 𝑡 + Δ𝑡) − 𝑓𝑖 (𝑥, 𝑡)
1
𝜔𝑖
= [𝑓𝑖𝑒𝑞 ( 𝜌(𝑥, 𝑡), 𝑢
⃗ (𝑥, 𝜌, 𝑡)) − 𝑓𝑖 (𝑥, 𝑡)] − 3 2 Δ𝑡𝑒𝑖 ∙ ∇𝑃
𝜏
𝑐
(11)
𝑓𝑖 is the particle distribution function, and 𝑓𝑖𝑒𝑞 is the equilibrium distribution function.
𝑓𝑖𝑒𝑞 (𝜌, 𝑢
⃗ ) = 𝜌𝜔𝑖 [1 +
3
9
3
𝑒𝑖 ∙ 𝑢
⃗ + 4 (𝑒𝑖 ∙ 𝑢
⃗ )2 − 2 𝑢
⃗ ∙𝑢
⃗]
2
𝑐
𝑐
2𝑐
(12)
i(=0,1,…,N-1) denotes the discrete directions and N is the number of directions of the
LBM. 𝑒𝑖 is the directional vector of ith direction. The weights, 𝜔𝑖 =1/3 for i=0, 𝜔𝑖 =1/18
for i=1,…,6, and 𝜔𝑖 =1/36 for i=7,…,18 when the number of directions, N is 19. 𝜏 is the
single relaxation parameter and it is related with the viscosity of the fluid. 𝜈 =
(2𝜏 − 1)𝑐 2 Δ𝑡/6. The flow is in the regime of the creeping flows (Stokes’ flows) which
can be also obtained by the Stokes’ equation. The Stokes’ equation has no dimensionless
parameters in the dimensionless form of the Stokes’ equation, and so, the dimensionless
solution is not changed by the change of the viscosity[26]. We can choose any value for
the relaxation parameter, and 𝜏=0.75 is used.
24
Figure 4. Discrete velocity directions of LBM. N=19[27].
Furthermore, the LBM does not require any extra spatial discretization while the
FDM, FVM, and FEM requires to discretisize the computational domain first. The step to
construct a grid system of an irregular porous media needs a huge effort[30]. We skip the
step with the LBM because the segmented data are directly used as the lattice of the LBM
computation. For each channel, we just select a proper region around the corresponding
throat for the 3-dimensional geometry of the channel. We find all void voxels within a
distance of 6 voxels from the throat by burning out the throat barrier voxels and assign
one end as an inlet and the other as outlet. The figure is one example of computational
domains for the Lattice Boltzmann method. Volume flow rate is obtained from the flow
solution and the channel conductance is also calculated directly from the solution.
25
Figure 5. An example of the computation domain for the LBM. Throat, inlet, and outlet
voxels are shaded grey[27].
26
4. Network flow model for CO2 sequestration
a. The equation of the network flow model
We simulate the process of CO2 sequestration in a sample core. Saline water with
CO2 resolved is injected into the sample core and the saline water is transported through
the porous rock interacting with accessible minerals. The saline water is highly acidic
because CO2 is resolved under very high pressure. The simulation includes transport of
saline water and some aqueous species and some chemical reactions between some
species and minerals and among species. We use the single-phase network flow model to
compute volume flow rates through all channels for the simulation.
Figure 6. A schematic diagram of the simulation of CO2 sequestration. CO2 resolved water is
injected from the top boundary into the porous rock. A flow is induced downward
chemically interacting with minerals in the rock.
27
The below ordinary differential equation is the equation of the network flow
model and it shows advection of species by volume flow rates, diffusion of species, and
kinetic reactions of species with minerals[4].
For each pore i,
𝑉𝑖
[∙]j − [∙]i
𝑑[∙]𝑖
+ ∑ 𝑄𝑖𝑗 [∙]𝑖 + ∑ 𝑄𝑖𝑗 [∙]𝑗 = 𝑆𝑖,[∙] + ∑ 𝐷[∙] 𝑎𝑖𝑗
𝑑𝑡
𝑙𝑖𝑗
𝑄𝑖𝑗 >0
𝑗
:
𝑄𝑖𝑗 <0
index of a neighboring pore.
𝑉𝑖
:
pore volume of pore i.
[∙]𝑖
:
concentrations of species.
𝑄𝑖𝑗
:
volume flow rate from pore i to pore j.
𝐷[∙]
:
diffusion coefficients of species ∙
𝑎𝑖𝑗
:
throat area between pore i and pore j.
𝑆𝑖,[∙]
(13)
𝑗
:
kinetic reaction source term of the species ∙ in pore i.
𝑆𝑖,[∙] = (𝑟𝐴 𝐴𝐴 + 𝑟𝐾 𝐴𝐾 )[∙]
𝑟𝐴 , 𝑟𝐾 : anorthite and kaolinite reaction rates of pore i.
𝐴𝐴 , 𝐴𝐾 : surface areas of anorthite and kaolinite in pore i.
The equation of the network flow model can be solved numerically by the Euler
method or 4th order Runge-Kutta method[31].
28
b. Kinetic reactions
In the above equation of the network flow model, the term 𝑆𝑖,[∙] = (𝑟𝐴 𝐴𝐴 +
𝑟𝐾 𝐴𝐾 )[∙] represents two kinetic reactions, anorthite reaction and kaolinite reaction[4; 32].
The anorthite reaction is
CaAl2Si2O8(s)+8H+↔Ca2++2Al3++2H4SiO4
and the kaolinite reaction is
Al2Si2O5(OH)4(s)+6H+↔2Al3++2H4SiO4+H2O.
The reaction rates, 𝑟𝐴 , 𝑟𝐾 are modeled based on some experiments. The reaction rate of
anorthite is
𝑟𝐴 = (𝑘𝐻 {𝐻 + }1.5 + 𝑘𝐻2 𝑂 + 𝑘𝑂𝐻 {𝑂𝐻}0.33)(1 − ΩA )
Ω𝐴 =
{𝐶𝑎2+ }{𝐴𝑙 3+ }2 {𝐻4 𝑆𝑖𝑂4 }2
{𝐻 + }8 𝐾𝑒𝑞,𝐴
(14)
(15)
and the reaction rate of kaolinite is
𝑟𝐾 = (𝑘𝐻 {𝐻 + }0.4 + 𝑘𝑂𝐻 {𝑂𝐻}0.3 )(1 − Ω0.9
K )
(16)
{𝐴𝑙 3+ }2 {𝐻4 𝑆𝑖𝑂4 }2
Ω𝐾 =
{𝐻 + }6 𝐾𝑒𝑞,𝐾
(17)
29
Ω is the saturation state and it indicates over- or under-saturation. A reaction is
dissolution when Ω < 1. It is precipitation when Ω > 1. {∙} is the activity of a species
and we will discuss the activity in the later section 4.f. The reaction rate constants of
anorthite and kaolinite at 25oC are listed in the below Table 1, and the equilibrium
constants of anorthite and kaolinite at 50oC are listed too. The simulation temperature is
50oC and so, the reaction rates are adjusted by the Arrhenius equation[33].
1
1
𝑘
−𝐸 ( −
)
= 𝑒 𝐴𝑟 𝑅𝑇 𝑅𝑇0
𝑘0
(18)
The adjusted reaction rates at 50oC are also listed in the Table 1.
Table 1. Equilibrium and reaction rate constants of anorthite and kaolinite reactions
log k (mol/m2s)
o
log Keq (50 C)
Anorthite
Kaolinite
Ea(kJ/mol)
log kH
log kH2O
log kOH
25oC
-3.32
-11.6
-13.5
50oC
-3.07
-11.35
-13.25
25oC
-10.8
-
-15.7
21.7
-18.4
3.80
-29.3
o
50 C
-10.40
30
-
-15.30
c. Equilibrium computation of instantaneous reactions
The simulation considers water and 14 aqueous species, H2O, H+, OH-, H2CO3,
HCO3-, CO32-, H4SiO4, H3SiO4-, H2SiO42-, Al3+, Al(OH)2+, Al(OH)2+, Al(OH)3, Al(OH)4-,
and Ca2+. There are important 9 chemical reactions for the simulation, and the 15 species
always follows the 9 reactions instantaneously[4]. We need to compute the equilibrium
state of the 9 instantaneous reactions. The simulation first, computes the equation of the
network flow model to apply the advection, diffusion of all 15 species, and the two
kinetic reactions of anorthite and kaolinite during the time step, Δt. The equilibrium
computation of the instantaneous reactions follows, and the equilibrium occurs without
any change of time. The Figure 7 depicts the above simulation procedure.
Equilibrium computation
*no time advance
t
t+Δt
Network flow model
• kinetic reaction
• advection
• diffusion
*time advance by Δt
Figure 7. One time step in the simulation
The Table 2 shows the 9 instantaneous reactions, and the corresponding
equilibrium constants[4; 34]. The first step of the equilibrium computation is the selection
of components from the 15 species[33]. There are 15 species and 9 reactions, and thus, 6
31
species should be used for the set of components. There are some appropriate choices of
components, and one choice is { H2O, H+, H2CO3, H4SiO4, Al3+, and Ca2+ }. Then, the
activities of the other species can be expressed by the activities of the 6 selected
components, and the Table 3 shows the 9 derived equilibrium equations of the
instantaneous reactions to compute the activities. The left hand side of each reaction is
one of the other 9 species, and the right hand side is composed of a few components. {∙}
represent the activity of a species and [∙] represent the concentration of a species. The
Ca2+ appears in the simulation by the kinetic reactions but there is no instantaneous
reaction of Ca2+. By the equilibrium of the instantaneous reactions, [Ca2+] remains the
same.
Table 2. Instantaneous reactions and equilibrium constants of the reactions
Reactions
logKeq
1.
H2O⇔H++OH-
-13.2
2.
H2CO3*⇔HCO3-+H+
-6.15
3.
HCO3-⇔CO32-+H+
-10.0
4.
H4SiO4⇔H3SiO4-+H+
-9.2
5.
H3SiO4-⇔H2SiO42-+H+
-12.4
6.
Al3++ OH- ⇔Al(OH)++
8.76
7.
Al3++2OH-⇔Al(OH)2+
18.9
8.
Al3++3OH-⇔Al(OH)3
27.3
9.
Al3++4OH-⇔Al(OH)4-
33.2
32
Table 3. 9 derived instantaneous reaction, and corresponding equilibrium equations to
compute the activities of the 9 non-component species.
Reactions
logKeq
1.
H2O – H+ ⇔ OH-
-13.2
{OH − } = K1 {H2 O}{H+ }−1
2.
H2CO3* – H+ ⇔ HCO3-
-6.15
∗
+ −1
{HCO−
3 } = K 2 {H2 CO3 }{H }
3.
H2CO3* – 2H+ ⇔ CO32-
-16.15
∗
+ −2
{CO2−
3 } = K 3 {H2 CO3 }{H }
4.
H4SiO4 – H+ ⇔ H3SiO4-
-9.2
+ −1
{H3 SiO−
4 } = K 4 {H4 SiO4 }{H }
5.
H4SiO4 – 2H+ ⇔ H2SiO42-
-21.6
+ −2
{H2 SiO2−
4 } = K 5 {H4 SiO4 }{H }
6.
Al3+ + H2O – H+ ⇔ Al(OH)++
-4.44
{Al(OH)++ } = K 6 {Al3+ }{H2 O}{H + }−1
7.
Al3+ + 2H2O – 2H+ ⇔ Al(OH)2+
-7.5
3+
2
+ −2
{Al(OH)+
2 } = K 7 {Al }{H2 O} {H }
8.
Al3+ + 3H2O – 3H+ ⇔ Al(OH)3
-12.3
{Al(OH)3 } = K 8 {Al3+ }{H2 O}3 {H + }−3
9.
Al3+ + 4H2O – 4H+ ⇔ Al(OH)4-
-19.6
3+
4
+ −4
{Al(OH)−
4 } = K 9 {Al }{H2 O} {H }
From the equation in the Table 3, we know all activities when the activities of the
components are given. The activity is different from the concentration and it is the real
effective reactivity of a species. We use the activity coefficient to express the activity, and
the activity coefficient is defined as the ratio of the activity to the concentration of the
species,
𝛾=
{∙}
[∙]
We will discuss how to compute the activity coefficient and the activity later.
33
(19)
The equilibrium state satisfies the mass balance of the species. Masses of species
are not conserved, but masses at equilibrium are balanced by the chemical reactions. We
use new variables, the total concentrations to apply the mass balance of 15 species. For
example, the total concentration of H2CO3* is defined by TOT[H2CO3*] =
[H2CO3*]+[HCO3-]+[CO32-]. Let the subscript,
0
species out of equilibrium and the subscript,
denote the concentrations of the given
species in
equilibrium.
Then,
eq
denote given concentrations of the
TOT[H2CO3*]0=[H2CO3*]0+[HCO3-]0+[CO32-]0
and
TOT[H2CO3*]eq=[H2CO3*]eq+[HCO3-]eq+[CO32-]eq. While [HCO3-] increases from [HCO3]0 to [HCO3-]eq, the same amount of [H2CO3*] is consumed by the instantaneous reaction
#2 in the Table 3. If [HCO3-] decreases, then the same amount of [H2CO3*] is produced.
Likewise, the change of [CO32-] induces the change of [H2CO3*] in the opposite direction
by the instantaneous reaction #3 in the Table 3. Thus, TOT[H2CO3*] is conserved, i.e.,
TOT[H2CO3*]0= TOT[H2CO3*]eq. Moreover, we can define the total concentrations of all
components, and the total concentrations are conserved in equilibrium, i.e.,
TOT[∙]0=TOT[∙]eq for all components[33]. The total concentration of all components are
listed in the below Table 4.
Table 4. Total concentrations of all components.
components
TOT[∙]
H2O
[H2O]+[OH-]+[Al(OH)++]+2[Al(OH)2+]+3[Al(OH)3]+4[Al(OH)4-] ≃ [H2O]
[H+]−[OH-]−[HCO3-]−[CO32-]−[H3SiO4-]−[H2SiO42-]
+
H
−[Al(OH)++]−[Al(OH)2+]−[Al(OH)3]−[Al(OH)4-]
H2CO3*
[H2CO3*]+[HCO3-]+[CO32-]
34
H4SiO4
[H4SiO4]+[ H3SiO4-]+[ H2SiO42-]
Al3+
[Al3+]+[ Al(OH)++]+[ Al(OH)2+]+[ Al(OH)3] +[Al(OH)4-]
Ca2+
[Ca2+]
Now, we construct the Table 5, which has whole information for the equilibrium
computation of the 9 instantaneous reactions. The first 6 rows of the table are for the 6
components and the other 9 rows, which have the exponents and the equilibrium
constants of the reactions to compute the activities of the 9 species other than components.
The columns of components show the coefficients of the species for the total
concentrations.
35
Table 5. Table for the equilibrium computation.
D
log
H2O
H+
H2CO3*
H4SiO4
Al3+
Ca2+
Reaction
Molar
z
(10-10
Keq
mass
m2/s)
1. H2O
1
0
0
0
0
0
-
0.0
0
-
18.0152
2. H+
0
1
0
0
0
0
-
0.0
+1
93.1
1.0079
3. H2CO3*
0
0
1
0
0
0
-
0.0
0
19.2
62.0252
4. H4SiO4
0
0
0
1
0
0
-
0.0
0
17.0
96.1152
5. Al3+
0
0
0
0
1
0
-
0.0
+3
5.59
26.9815
6. Ca2+
0
0
0
0
0
1
-
0.0
+2
7.93
40.080
7. OH-
1
-1
0
0
0
0
H2O – H+ ⇔ OH-
-13.2
-1
52.7
17.0073
8. HCO3-
0
-1
1
0
0
0
H2CO3* – H+ ⇔ HCO3-
-6.15
-1
11.8
61.0173
9. CO32-
0
-2
1
0
0
0
H2CO3* – 2H+ ⇔ CO32-
-16.15
-2
9.65
60.0094
10. H3SiO4-
0
-1
0
1
0
0
H4SiO4 – H+ ⇔ H3SiO4-
-9.2
-1
12
95.1073
11. H2SiO42-
0
-2
0
1
0
0
H4SiO4 – 2H+ ⇔ H2SiO42-
-21.6
-2
8
94.0994
12. Al(OH)++
1
-1
0
0
1
0
Al3+ + H2O – H+ ⇔ Al(OH)++
-4.44
+2
8
43.9888
13. Al(OH)2+
2
-2
0
0
1
0
Al3+ + 2H2O – 2H+ ⇔ Al(OH)2+
-7.5
+1
12
60.9961
14. Al(OH)3
3
-3
0
0
1
0
Al3+ + 3H2O – 3H+ ⇔ Al(OH)3
-12.3
0
15
78.0034
15. Al(OH)4-
4
-4
0
0
1
0
Al3+ + 4H2O – 4H+ ⇔ Al(OH)4-
-19.6
-1
10
95.0107
36
Now, we can compute the all activities if the activities of the 6 components are
given. Thus, the problem of equilibrium computation is how to find the activities of 6
components which satisfy the mass balance, i.e. the total concentration conservation.
When we guess a set of activities of the components, we can compute the difference,
ΔTOT[∙]gs=TOT[∙]gs−TOT[∙]0 between the total concentrations of the guessed activities of
components and the total concentrations of the given concentrations. We can make a
function of which input and output are respectively, the set of the guessed activities of the
6 components and the set of the difference, ΔTOT[∙]gs=TOT[∙]gs−TOT[∙]0. Let F() be the
function
F( {H2 O}gs , {H + }gs , {H2 CO3∗ }gs , {H4 SiO4 }gs , {Al3+ }gs , {Ca2+ }gs )
= ( Δ𝑇𝑂𝑇[H2 O]gs , Δ𝑇𝑂𝑇[H + ]gs , Δ𝑇𝑂𝑇[H2 CO3∗ ]gs , Δ𝑇𝑂𝑇[H4 SiO4 ]gs ,
(20)
Δ𝑇𝑂𝑇[Al3+ ]gs , Δ𝑇𝑂𝑇[Ca2+ ]gs ) .
At the equilibrium concentrations, the function,
F( {H2 O}eq , {H + }eq , {H2 CO∗3 }eq , {H4 SiO4 }eq , {Al3+ }eq , {Ca2+ }eq ) = ( 0, 0, 0, 0, 0, 0 )
because the total concentrations, TOT[∙] is conserved in the equilibrium computation, i.e.,
ΔTOT[∙]=0, and TOT[∙]0=TOT[∙]eq. The computation of the equilibrium state is
numerically a root finding of a nonlinear 6-variable function, F(). We use the NewtonRaphson method to find the solution, the set of the component activities in the
equilibrium state. The Newton-Raphson method is one of the basic methods to find a root
numerically[35]. Let F:Rn→ Rn and xk∈Rn be the k-th approximation for the root. The
next approximation is estimated by 𝑥𝑘+1 = 𝑥𝑘 − 𝐽𝑥𝑛 −1 𝐹(𝑥𝑘 ) where 𝐽𝑥𝑛 is the Jacobian
matrix at xn.
37
The water activity is dimensionless 1.0, and water concentration is constant. Thus,
the activity and concentration is not a variable in the previous equilibrium computation.
In the actual computation, water is not considered, but for convenience, we keep the
water in the equilibrium computation.
There is another way to compute the equilibrium state. In the Table 5, H+ is
coupled with the all non-component species but the other components are not coupled
with each other. Therefore, we can compute the activities of all the other components
from the estimated {H+}gs and the total concentrations, TOT[∙]0. For example, consider
the total concentrations of H2CO3* and the corresponding species, H2CO3*, HCO3-, and
CO32-. The equation of the total concentration of H2CO3* is converted to obtain {H2CO3*}.
𝑇𝑂𝑇[H2 CO3∗ ]0 = [H2 CO3∗ ] + [HCO3− ] + [CO2−
3 ]
=
{H2 CO3∗ } {HCO−
{CO2−
3}
3 }
+
+
γH2 CO∗3
γHCO−3
γCO2−
3
{H2 CO3∗ } K 2 {H2 CO∗3 }{H + }−1 K 3 {H2 CO3∗ }{H + }−2
=
+
+
γH2 CO∗3
γHCO−3
γCO2−
3
1
K 2 {H + }−1 K 3 {H + }−2
= {H2 CO3∗ } (
+
+
)
γH2 CO∗3
γHCO−3
γCO2−
3
{H2 CO3∗ }
=
𝑇𝑂𝑇[H2 CO3∗ ]0 (
(21)
−1
K 2 {H + }−1 K 3 {H + }−2
+
+
)
γH2 CO∗3
γHCO−3
γCO2−
3
1
From the above equation, {H2CO3*} is determined by the {H+} and TOT[H2CO3*] and
moreover, the {H2CO3*} obtained by the above equation satisfies the conservation of
TOT[H2CO3*]. Likewise, the activities of the other components are computed by the
below equations.
38
−1
K 4 {H + }−1 K 5 {H + }−2
{H4 SiO4 } = 𝑇𝑂𝑇[H4 SiO4 ]0 (
+
+
)
γH4 SiO4
γH3 SiO−4
γH2 SiO2−
4
1
{Al3+ }
K 6 {H + }−1 K 7 {H + }−2 K 8 {H + }−3
+
+
+
0(
γAl3+ γAl(OH)++
γAl(OH)2 +
γAl(OH)3
3+ ]
= 𝑇𝑂𝑇[Al
(22)
1
(23)
−1
K 9 {H + }−4
+
)
γAl(OH)−1
4
{Ca2+ } = 𝑇𝑂𝑇[Ca2+ ]0 γCa2+
(24)
By the above equations, we have the activities of all components which satisfy the
conservation of TOT[H2CO3*], TOT[H4SiO4], TOT[Al3+], TOT[Ca2+], and TOT[H2O]
when {H+} is given. The only one condition unsatisfied is the conservation of TOT[H+].
For a given {H+}gs, we derive the activities of the other components by the above
equations, and we compute TOT[H+]gs and ΔTOT[H+]. Thus, the problem of the
equilibrium computation is how to find {H+} to satisfy ΔTOT[H+]=0, i.e.,
TOT[H+]=TOT[H+]0. We express the above procedure by a function, f:R→ R such that
f( [H+]gs )=( ΔTOT[H+] ) and f( [H+]eq ) = 0. We can use the bisection or the secant
method to find the root of f( ).
This new method is more simple and faster than the previous method, and
furthermore, the bisection method is stable, i.e., it never fails to find the equilibrium state.
The new method uses the simplicity of the system of the reactions, but it cannot be used
39
for a general system of chemical reactions. For example, if there is a reaction which has
H2CO3* and H4SiO4 together, the simple method is not applicable any more.
d. CO2 solubility in aqueous NaCl solution
The injected inflow from the top boundary into the porous rock has some amount
of CO2 resolved and an experimental method is introduced[36]. The CO2 solubility is
determined from the balanced between the chemical potentials of CO2 in the liquid phase,
𝑙
𝑣
μlCO2 and in the gas phase, μvCO2 . At equilibrium, 𝜇𝐶𝑂
= 𝜇𝐶𝑂
, and the expansion of
2
2
excess Gibbs energy is applied.
𝑙(0)
𝑦𝐶𝑂2 𝑃 𝜇𝐶𝑂2
ln
=
− ln 𝜙𝐶𝑂2 (𝑇, 𝑃, 𝑦) + ∑ 2𝜆𝐶𝑂2 −𝑐 𝑚𝑐
𝑚𝐶𝑂2
𝑅𝑇
𝑐
(25)
+ ∑ 2𝜆𝐶𝑂2 −𝑎 𝑚𝑎 + ∑ ∑ 𝜁𝐶𝑂2 −𝑐−𝑎 𝑚𝑐 𝑚𝑎
𝑎
𝑐
𝑎
where the subscript, ‘c’ represents cations and ‘a’ represents anions. The fraction of CO2
vapor pressure, yCO2 is estimated by
𝑦𝐶𝑂2 =
𝑃 − 𝑃𝐻2 𝑂
𝑃
where the pure water pressure, PH2 O is obtained by
40
𝑃𝐻2𝑂 =
where t =
T−Tc
Tc
and 647.29K.
𝑃𝑐 𝑇
[1 + 𝑐1 (−𝑡)1.9 + 𝑐2 𝑡 + 𝑐3 𝑡 2 + 𝑐4 𝑡 3 + 𝑐5 𝑡 4 ]
𝑇𝑐
and the critical pressure, Pc and critical temperature, Tc are 220.85bar
l(0)
2
μCO
RT
, λCO2 , and ζCO2 are computed by the below equation.
𝑋(𝑇, 𝑃) = 𝑐1 + 𝑐2 𝑇 +
𝑐3
𝑐5
+ 𝑐4 𝑇 2 +
+ 𝑐6 𝑃 + 𝑐7 𝑃 𝑙𝑛 𝑇
𝑇
630 − 𝑇
𝑃
𝑃
𝑐10 𝑃2
+ 𝑐8 + 𝑐9
+
𝑇
630 − 𝑇 (630 − 𝑇)2
The coefficients, ci’s for
l(0)
2
μCO
RT
(26)
, λCO2 , and ζCO2 are listed in the table. The fugacity
coefficient of CO2, 𝑙𝑛 𝜙 (𝑇, 𝑃) is computed by the equation.
𝑎1 +
𝑙𝑛 𝜙(𝑇, 𝑃) = 𝑍 − 1 − 𝑙𝑛 𝑍 +
𝑎7 +
+
+
𝑎2 𝑎3
𝑎
𝑎
+
𝑎4 + 52 + 63
𝑇𝑟2 𝑇𝑟3
𝑇𝑟 𝑇𝑟
+
𝑉𝑟
2𝑉𝑟
𝑎8 𝑎9
𝑎
𝑎
+ 3 𝑎10 + 11
+ 12
2
2
𝑇𝑟 𝑇𝑟
𝑇𝑟
𝑇𝑟3
+
4𝑉𝑟
5𝑉𝑟
𝑎13
𝑎15
𝑎15
[𝑎
+
1
−
(𝑎
+
1
+
)
𝑒𝑥𝑝
(−
)]
14
14
𝑉𝑟2
𝑉𝑟2
2𝑇𝑟3 𝑎15
where
41
(27)
𝑃𝑟 𝑉𝑟
𝑍=
=1+
𝑇𝑟
𝑎1 +
𝑎2 𝑎3
𝑎
𝑎
𝑎
𝑎
+ 3 𝑎4 + 52 + 63 𝑎7 + 82 + 93
2
𝑇𝑟 𝑇𝑟
𝑇𝑟 𝑇𝑟
𝑇𝑟 𝑇𝑟
+
+
2
𝑉𝑟
𝑉𝑟
𝑉𝑟4
𝑎
𝑎
𝑎10 + 11
+ 12
2
𝑎13
𝑎15
𝑎15
𝑇𝑟
𝑇𝑟3
+
+ 3 2 (𝑎14 + 2 ) 𝑒𝑥𝑝 (− 2 )
5
𝑉𝑟
𝑉𝑟
𝑇𝑟 𝑉𝑟
𝑉𝑟
𝑃𝑟 =
𝑃
,
𝑃𝑐
𝑇𝑟 =
𝑇
,
𝑇𝑐
𝑉𝑟 =
(28)
𝑉
𝑉𝑐
The critical pressure, Pc=220.85 and the critical temperature, Tc=647.29K. Vc is defined
by
𝑉𝑐 =
𝑅𝑇𝑐
𝑃𝑐
R is the universal gas constant, and R=8.314467 Pa m3/K∙mol.
e. Initial and boundary conditions
The domain of the simulation is the pore space of a rock. From the top boundary,
the acidic saline water with CO2 resolved is injected into the porous rock. The pressure of
CO2 is given and the amount of CO2 resolved in the inflowing water is calculated by the
method explained in the previous section[4; 36]. [H2CO3*] of the inflow is the CO2
solubility and the state of the inflow is the equilibrium state of the 9 instantaneous
reactions with the computed [H2CO3*]. There is no silica-bearing species or aluminumbearing species in the computation of the state of the inflowing saline. We add Ca2+ to the
42
inflow to consider the equilibrium of the saline water with quartz and kaolinite, and [Ca2+]
=7.9×10-6M in the inflow from the top boundary. The saline has the total dissolution
(TDS), 0.45M. Thus, [Na+]=[Cl-]=0.45M[4].
The water flows out through the bottom boundary and no flow comes in the
porous rock from the bottom boundary. The side boundaries are sealed, i.e., no flow
through the side boundaries.
Initially, the pore space of the rock is filled with saline water, and the water is in
equilibrium of the 9 instantaneous reactions. Similarly, [Ca2+]=7.9×10-6M, and [Na+]=[Cl]=0.45M[4].
f. Activity coefficients and diffusion coefficients
The activity is different from the concentration and it is the real effective
reactivity of a species. We use the activity coefficient to express the activity. The activity
coefficient is the ratio of the activity to the concentration of the species,
𝛾=
{∙}
[∙]
(29)
The activity coefficients of aqueous species can be estimated by the Davis’ equation[33].
First, we define the ionic strength, I.
𝐼=
1
∑ m𝑖 z𝑖2 𝑉𝑖
2
𝑖
43
(30)
where mi is the molar concentration of species i in M (mol/liter) and zi is the ionic charge
of the species i. The activity coefficient, γ is computed by the Davis’ equation for I<0.5M.
log 𝛾 = −Az 2 (
𝐼1/2
− 0.2𝐼)
1.0 + 𝐼1/2
(31)
where A = 1.82 × 106 (εT)−3/2 and ε is the dielectric constant. A=0.51 for water at
25oC. For the value of A at 50oC, we use the below table of the dielectric constants of
pure water, ε, at certain temperatures[37].
Table 6. Table of temperature vs. dielectric constant of pure water.
T
ε
0oC
87.7
25oC
78.3
50oC
69.9
100oC
55.7
The water activity is close to dimensionless 1.0 under moderate environment,
and the water activity of dimensionless 1.0 is accurate enough for many analyses in water
chemistry. We use the model of the saline water activity, and the water activity by this
model is also 1.0[37]. The water activity and water concentration is 1.0 and 54.7M
respectively in the simulation of CO2 sequestration. Thus, there are only 4 unknown
activities/concentrations out of 6 components in the equilibrium computation since {H2O}
=1.0, [H2O]=54.7M, and [Ca2+] are not changed.
44
The equation of the network flow model has the diffusion term, ∑𝑗 𝐷[∙] 𝑎𝑖𝑗
[∙]j −[∙]i
𝑙𝑖𝑗
,
and we need the diffusion coefficients of all species. The value of the diffusion
coefficients are listed in the previous Table 5, and all diffusion coefficients are under
T=25oC and P=101.3kPa[37; 38]. The simulation of CO2 sequestration is under T=50o
and P=100bar, and the diffusion coefficients are adjusted to those under the simulation
condition by the Stokes-Einstein relation[37].
𝐷𝑇1 × 𝜇𝑤,𝑇1 𝑇1 = 𝐷𝑇2 × 𝜇𝑤,𝑇2 𝑇2
(32)
T1 and T2 are two given temperatures, 𝐷𝑇1 and 𝐷𝑇2 are the diffusion coefficients at T1
and T2, and 𝜇𝑤,𝑇1 and 𝜇𝑤,𝑇2 are water viscosities at T1 and T2.
g. Effective and volume-averaged reaction rates
From the network flow model, the total mass change rates of anorthite and
kaolinite can be defined by the sum of the mass change rate of all pores. The total mass
change rates are the rate of consumed (produced) mass of anorthite and kaolinite, and
they are simply computed by the equation. The unit of the mass change rates is mol/s.
𝑀𝐴 = ∑ 𝐴𝐴,𝑖 𝑟𝐴,𝑖
𝑖=1
𝑀𝐾 = ∑ 𝐴𝐾,𝑖 𝑟𝐾,𝑖
𝑖=1
45
(33)
The subscripts, A and K represent anorthite and kaolinite respectively, and A and r are the
mineral surface area and kinetic reaction rate of a pore. The effective reaction rates are
defined by the below equation. The unit is mol/m2s.
𝑅𝑁,𝐴 =
𝑅𝑁,𝐾
∑𝑖=1 𝐴𝐴,𝑖 𝑟𝐴,𝑖
∑𝑖=1 𝐴𝐴,𝑖
∑𝑖=1 𝐴𝐾,𝑖 𝑟𝐾,𝑖
=
∑𝑖=1 𝐴𝐾,𝑖
(34)
The effective reaction rates are the true effective values of the reaction rates in the whole
network. The volume-averaged reaction rate is another reaction rate, which is computed
with the volume-averaged concentrations. The volume-averaged concentration is
obtained by dividing the total mass of the species in the whole network by the pore
volume of the whole network.
̅̅̅ =
[∙]
∑𝑖=1[∙]𝑖 𝑉𝑖
∑𝑖=1 𝑉𝑖
(35)
̅̅̅
The volume-averaged activity can be defined by ̅̅̅
[∙] = γ̅̅̅
[∙] {∙}, and the volume-averaged
reaction rate is defined by the equations (14), (15) of the reaction rates with the volumeaveraged activities.
+ }1.5 + 𝑘
− 0.33 )(1 − ̅
̅̅̅̅̅̅
̅̅̅̅̅̅̅̅̅
̅̅̅
𝑅𝐴 = (𝑘𝐻 {𝐻
𝛺̅̅𝐴̅)
𝐻2 𝑂 + 𝑘𝑂𝐻 {𝑂𝐻 }
+ }0.4 + 𝑘
+ 0.3
̅̅̅̅̅̅
̅̅̅̅̅̅̅̅̅
̅̅̅̅0.9
̅̅̅̅
𝑅𝐾 = (𝑘𝐻 {𝐻
𝑂𝐻 {𝑂𝐻 } ) (1 − 𝛺𝐾 )
(36)
̅̅̅̅
𝛺𝐴 and ̅̅̅̅
𝛺𝐾 are the saturation states of the anorthite and kaolinite from the volume46
averaged activities, ̅̅̅
{∙}. To compare the values of the two different reaction rates, the
ratio of the two reaction rates are defined[4].
𝛽𝐴 =
𝛽𝐾 =
̅̅̅
𝑅𝐴
𝑅𝑁,𝐴
̅̅̅̅
𝑅𝐾
𝑅𝑁,𝐾
47
(37)
III.
Computational results
1. Pore-throat network of mineral distributions
Three Viking samples, Viking3W4, Viking10W4, and Viking 14w5 which are
described as sandstone, shaly sandstone, and conglomerate sandstone respectively are
analyzed to construct the pore-throat networks of mineral distributions. The three Viking
samples are real rock samples from the abandoned oil/gas reservoirs in the Viking
formation in the Alberta basin. The samples are imaged at the X2B beam line at the
National Synchrotron Light Source at Brookhaven National Laboratory. The resolution of
the CT images is 3.98μm and the size is originally 723×723×600. The images are resized
to remove voxels of bad quality, and the sizes of resized Viking3W4, Viking10W4, and
Viking14W5 are 723×723×356, 723×723×405, and 723×723×405 respectively. The
corresponding real dimensions of the resized CT images are 2.88mm×2.88mm×1.42mm,
2.88mm×2.88mm×1.61mm, and 2.88mm×2.88mm×1.61mm respectively. The three pairs
of thresholds, T0 and T1 for the kriging based segmentations are listed in the Table 7 and
the histograms of the attenuation coefficients with the threshold windows are shown in
the Figure 8. The next Figure 9 (a) is one slice of the CT image of Viking3W4 and the rest
are three bi-phase segmented images for the four-phase image. The Figure 9 (b) is one
slice of the primary pore-primary grain segmented image, the (c) is one slice of the voidkaolinite segmented image in the primary pore space, and the (d) is one slice of the
quartz-the minerals of interest (MoI) segmented image in the primary grain space. The
next Figure 10 is the final four-phase image from the three bi-phase images after the
48
adjustment of the inter-phase surface of void space and MoI. In the images, the four
colors − black, green, grey, and red − indicate the four phases of void, kaolinite, quartz,
and the minerals of interest respectively. To compare the segmented images with the BSE
analysis, the mineral abundances and accessibilities of the CT images and BSE images
are listed in the Table 8. The correction step of the segmented images recovers the surface
area between the void space and the minerals of interest. The accessibilities expressed by
surface areas are corrected by the surface area to volume ratio, 40,000cm-1 from the BET
measurement, and the correction factors of the sandstone, the shaly sandstone, and the
conglomerate sandstone are 41.7, 34.2, and 61.5 respectively.
Table 7. Threshholds, T0 and T1 for the bi-phase segmentations.
Void
Sandstone
kaolinite
quartz
MoI
15150 - 15800
15950
- 16500
18100
-
18700
9150 - 9750
9900
- 10500
12550
-
13150
7700 - 8200
8400
-
11250
-
11950
Shaly
sandstone
Conglomerate
sandstone
49
9350
Figure 8. Histograms of the attenuation coefficients of three Viking samples with the
threshold windows.
50
(a)
(b)
(c)
(d)
Figure 9. (a) One slice of grey scale image of the conglomerate sandstone (Viking14W5). (b)
Bi-phase segmented image of the primary pore and primary grain spaces. Black and white
indicate the primary grain space and primary pore space respectively. (c) Segmented
image of kaolinite and void space in the primary pore space. Black and green indicate the
void space and kaolinite phase respectively. Grey indicates the primary grain space. (d)
Segmented image of quartz and MoI. Grey and red indicate quartz and MoI respectively.
White indicates the primary pore space.
51
Figure 10. The final four-phase segmented image of the conglomerate sandstone
(Viking14W5) after adjustment step. Black, green, grey, and red indicate void, kaolinite,
quartz, and MoI respectively.
Figure 11. Grey scale and four-phase segmented images of the sandstone (Viking3W4).
52
Figure 12. Grey scale and four-phase segmented images of the shaly sandstone
(Viking10W4).
Table 8. Mineral abundances and accessibilities of three Viking samples. The abundances
and accessibilities are compared with those by BSE analysis.
Sandstone
Shaly
sandstone
Mineral abundances(%)
Mineral accessibilities(%)
Porosity
kaoli.
quartz
MoI
kaoli.
quartz
MoI
BSE
18
21
73
6
81
17
2
Seg. only
13
14
84
2
64.1
35.8
0.06
corrected
14
15
82
3
63.4
35.3
1.3
BSE
7
31
64
5
86
13
1
Seg. only
4
10
89
1
70.2
29.8
0.03
corrected
4
10
89
1
69.8
29.5
0.7
BSE
8
5
94
1
65
35
<1
Seg. only
7
3
95
2
52.8
47.0
0.2
corrected
7
3
95
2
52.3
46.4
1.3
Conglom.
sandstone
The pore-throat network of the primary pore space is constructed and in the
Figure 13, the statistical distributions of pore volume, throat area, coordination number,
53
and medial axis length of the three rock samples are shown displaying the characteristics
of the three rocks. The sandstone has the largest number of pores and larger pore volumes
than the shaly sandstone because of the largest porosity of the sandstone. However, the
conglomerate sandstone, though it has the smallest porosity, has some large pore bodies.
In other words, the conglomerate sandstone has a small number of large pores. The
conglomerate sandstone has some big pores even though the porosity is small. This
overall tendency of the three rocks is also found consistently in the other statistical graphs
of throat area, coordination number, and channel lengths.
54
Figure 13. Statistics of the pore-throats networks of the primary pore space of the three
Viking samples. The sandstone data are represented by solid lines and ‘•’. The shaly
sandstone data are represented by dotted lines and ‘x’. The conglomerate sandstone data
55
are represented by dashed lines and ‘□’.
Mineral distributions of each pore are computed to construct the pore-throat
network of mineral distributions. The Figure 14 ~ Figure 19 are statistical graphs of the
volumes and surface areas of the kaolinite phase and the minerals of interest in a pore
body. There are a few new variables defined here. The percent volume of kaolinite of a
pore is defined as the percentage of the ratio of the kaolinite volume in the pore to the
pore volume. Likewise, the percent volume of MoI of a pore body is the percentage of the
ratio of the volume of MoI touched by the pore to the pore volume. In the previous Figure
2, the percent volumes of kaolinite and MoI are VG/Vpore and (VR1+VR2)/Vpore respectively.
The percent volume of kaolinite is between 0 and 100%, and the percent volume of MoI
can be more than 100%. The percent surface area of MoI of a pore is defined as the
percentage of the ratio of the surface area between MoI and the pore to the pore surface
area. In the Figure 2, it is (SRG+SRB)/Spore. There are two surface area of kaolinite in a
pore body. The first one is the surface area between kaolinite and void space, S GB in the
Figure 2, and the second one is the surface area between kaolinite and the primary grain
phase (quartz + MoI). The second surface area of the kaolinite is SGQ+SRG in the Figure 2.
The first and second percent surface areas of kaolinite are defined as the percentages of
the two areas divided by the pore surface area. The first percent surface area of kaolinite
can be greater than 100%, and the second percent surface area of kaolinite is between 0
and 100%. The Figure 14 has 6 graphs of the sandstone: the first graph has volume
distributions of pore, kaolinite, and MoI. The second graph has distributions of surface
areas of pore and MoI, and the second has the surface area of kaolinite. The third and
fourth graphs are the percent volumes of kaolinite and MoI. The last two graphs are the
56
percent surface areas of kaolinite and MoI. The Figure 15 and Figure 16 have the same
graphs of the shaly sandstone and conglomerate sandstone.
The number of pores which contain some kaolinite and the number of pores
which contain some MoI are in Table 9, and the means and standard deviations of the
volumes and surface areas of different phases are listed in the Table 10. The graphs and
tables show some interesting characteristics of the three samples. The shaly sandstone has
only 6% touching a cluster of the minerals of interest while the sandstone and
conglomerate sandstone have 20% and 24% pores respectively. The shaly sandstone has
the smallest means of the logarithm of the surface areas and the smallest means of the
pore volume and volume of the kaolinite. The conglomerate sandstone has the largest
mean of the logarithm of surface area of MoI though it has the smallest mean of
logarithm of volume of MoI. This means the conglomerate sandstone has large portion of
MoI. The reason of the above result is from the basic characteristics of the two rocks. The
graphs of volumes and surface areas show the same result, but the graphs of the percent
pore surface area of MoI show the opposite result. The conglomerate sandstone does not
seem to have a larger percent pore surface area of MoI than the others. This result can be
interpreted as the larger pore surface area of the conglomerate sandstone makes the
percent pore surface area small. The graphs of the percent pore surface area of the
minerals of interest of the conglomerate sandstone in the Figure 16 and Figure 19 show
the most of pores of the conglomerate sandstone have no more than 10% of the percent
pore surface area of MoI.
Table 9. the numbers of pores which contain kaolinite, or MoI.
sandstone
Shaly sandstone
57
Congl. sandstone
Number of pores
14589
11040
1971
Number of pores of Kaol.
14579
11037
1969
Number of pores of MoI
2855(20%)
655(6%)
464(24%)
Table 10. Mean and standard deviations of pores, kaolinite clusters, and MoI clusters.
sandstone
Shaly
Congl.
sandstone
sandstone
µlogV
σlogV
µlogV
σlogV
µlogV
σlogV
Pore
4.71
0.69
4.54
0.74
4.87
0.99
Kaol. in a pore
4.51
0.60
4.47
0.69
4.58
0.78
MoI touching a pore
4.14
1.08
3.82
0.95
3.61
0.89
Surf.
Pore
5.47
0.61
5.31
0.67
5.79
0.82
Area
Kaol. in a pore
4.96
0.72
4.71
0.82
5.47
0.82
(µm2)
MoI touching a pore
4.26
0.75
3.78
0.71
4.33
0.70
Volume
(µm3)
58
Figure 14. Statistics of the mineral abundances and accessibilities of the sandstone,
Viking3W4.
59
Figure 15. Statistics of the mineral abundances and accessibilities of the shaly sandstone,
Viking10w4.
60
Figure 16. Statistics of the mineral abundances and accessibilities of the conglomerate
sandstone, Viking14W5.
61
Figure 17. Percent pore volumes and percent pore surface areas of kaolinite and MoI.
Sandstone, Viking3W4.
62
Figure 18. Percent pore volumes and percent pore surface areas of kaolinite and MoI. Shaly
sandstone, Viking10w4.
63
Figure 19. Percent pore volumes and percent pore surface areas of kaolinite and MoI.
Conglomerate sandstone, Viking14w5.
64
2. Single-phase network flow model
The single-phase network flow model with channel conductance based on the
shape factor is applied to four Fontainebleau samples, FB7.5, FB13, FB18, and FB22.
The four samples are analyzed before to obtain basic properties of the pore structures[12].
The permeabilities of four Fontainebleau samples are computed with different weight
values, ω from 0 to 1 to choose a proper weight for further computations. The
computation result is shown in the Figure 22 and the computed permeabilities are
compared with the below experimental result of Fontainebleau sandstones[15; 39].
2.75 × 10−5 𝜙 7.33 ,
𝐾𝐵𝑍 = {
0.303𝜙 3.05 ,
for 𝜙 < 9
for 𝜙 ≥ 9
(38)
Different sandstones have different best values of ω. The best ω changes from 0.5 to 1.0,
and thus, ω=0.7 is used for the next single-phase network flow computation. The Figure
20 is the pore pressure distribution in the Fontainebleau 22% sample where the red color
indicates the maximum pressure, top boundary pressure and the blue color indicates the
minimum pressure, bottom boundary pressure. The next Figure 21 is the plot of height vs.
pore pressure. The two graphs of the pore pressures shows the pressure is distributed
linearly with respect to the height.
The single-phase network flow model with the Lattice Boltzmann conductance is
also used to obtain the permeability and the result is summarized in the Table 11 with the
previous result of the shape factor conductance. The permeability, KLB from the model of
the Lattice Boltzmann conductance is closer to the empirical permeability, KBZ of the
equation.
65
Table 11. Computation result of permeabilities, KLB, KSF, and KBZ[15].
Porosity
KLB
KSF, ω=0.7
KBZ
FB7.5
7.5%
50
55
71
FB13
13%
681
798
756
FB18
18%
2205
3715
2041
FB22
22%
3920
4697
3765
Figure 20. Pressure distribution. FB22. Ptop=101.3 and Pbot=0.
66
Figure 21. Pressure distribution. Ptop=101.3 and Pbot=0.
Figure 22. Permeabilities with respect to weight factor, ω[15].
67
3. Simulation of CO2 sequestration
a. Validation of the reactive models
To validate the developed code, a sample problem is applied and the result is
compared with a published result[4]. An artificial sample is constructed and the variables
of the network are generated by some random number generators following statistical
distribution from different analyses. The some values in the paper are not consistent. For
example, pore volume follows the log-normal distribution with μlog V = −3.42 and
σlog V = 0.51, and the generated mean is 6.6×10-4mm3, which is very different from the
theoretical mean, 7.6×10-4mm3 of the log-normal distribution. The following modified
process is applied to generate a network for the simulation validation.
The sample size is 5.0×1.7×5.0mm3 with porosity 14%. There are 9000 pores
regularly distributed; 30×10×30 in the x-, y-, and z-direction respectively. Each interior
pore is connected with six neighboring pores in the 6 directions, ±x, ±y, ±z. Pore volumes
are generated based on the log-normal distribution with the log-mean, μlog V = −3.42
and the log-standard deviation, σlog V = 0.51 in mm3. The theoretical arithmetic mean
σ2
of log-normal distribution is eμ+ 2 = 6.84 × 10−4 mm3 which makes the porosity
slightly greater than 14%. The total volume is adjusted by multiplying a factor so that the
porosity is 14%. Pore surface area follows the exponential distribution with μS =
6.0mm2 , and the surface area is regenerated if the surface area is less then Smin =
1
2
(4π)3 (3𝑉)3 which is the spherical surface area from the pore volume. For the channel
conductances, the log of channel conductances, 𝑋𝑖𝑗 = log 𝐶𝑖𝑗 are generated by the joint
Gaussian distribution with the sum, 𝑌𝑖𝑗 = log 𝑉𝑖 + log 𝑉𝑗 of log pore volumes of the two
connected pores. The mean and standard deviation of 𝑌𝑖𝑗 are 𝜇𝑌 = 2𝜇log 𝑉 and 𝜎𝑌 =
68
√2𝜎log 𝑉 and the mean and standard deviation of 𝑋𝑖𝑗 are 𝜇𝑋 = −6.1 and 𝜎𝑋 = 1.0 for
the absolute permeability, 80md. For the log conductance, 𝑋𝑖𝑗 of each channel, the mean
and standard deviation of 𝑋𝑖𝑗 of the channel is calculated by the below equation.
𝜇𝑋,𝑖𝑗 = 𝜇𝑋 + 𝜌
2
𝜎𝑋,𝑖𝑗
𝜎𝑋
(𝑌 − 𝜇𝑌 )
𝜎𝑌 𝑖𝑗
(39)
= 𝜎𝑋2 (1 − 𝜌2 )
2
For each channel, 𝜇𝑋,𝑖𝑗 and 𝜎𝑋,𝑖𝑗
are computed by the above equation, and 𝑋𝑖𝑗 of the
channel is generated by the normal distribution random number generator. The channel
conductance is obtained by 𝐶𝑖𝑗 = e 𝑋𝑖𝑗 . The channel cross section area is computed by the
Poiseuille’s flow and the channel conductance. The Poiseuille’s flow through a circular
𝜋𝑑4
𝜋𝑑4
pipe has the volume flow rate of 𝑄 = 128𝜇 (−𝛻𝑃) and so, the conductance is 𝐶 = 128𝜇.
From the equation, the diameter and the area can be computed and the area is used as the
channel cross section area. The channel lengths in the z-direction are simply computed by
dividing the sample height by the number of channels in the z-direction, and similarly, the
channel lengths in the other directions are computed. 900 pores are selected from the
9000 pores, and the 900 pores contain the reactive minerals, anorthite and kaolinite. A
block of 2×2×15 pores is selected randomly and all 60 pores in the block are assigned as
reactive pores. The 2×2×15 block is used to consider the size of feldspars. There can be
some overlapped pores between a pair of blocks and thus, a few extra small blocks of
pores are selected to obtain at least 900 reactive pores. The total reactive pores can be
slightly more than 900. The reactive surface area of anorthite and kaolinite are halves of
the reactive pore surface area. There can be a few pores which have extremely large value
69
of the ratio, 𝑆𝐴 /𝑉 of anorthite surface area to the pore volume because the surface area
is generated independently of the pore volume. This causes a huge concentration change
in the pore by the anorthite kinetic reaction, and restricts the time step size. To avoid this
problem, the pore volume of a pore body which has extremely large 𝑆𝐴 /𝑉 is increased to
a certain required minimum value. This process may increase the porosity by 0.01%. A
network is generated by the above process using random number generators of MATLAB,
and the variables of the network are summarized in the Table 12.
The initial state of all pores are the equilibrium state of saline with the total
dissolved salinity (TDS) 0.45M at T=50o and P=100bar. To consider the equilibrium with
mineral, [Ca2+]=7.9×10-6M is added to the initial state. The initial state is shown in the
Table 14. The saline water is injected from the top boundary, the water flows in the z-
direction, and it flows out though the bottom boundary. The water does not flow through
any side boundary. The solubility of CO2 to the water of top boundary is 2.0M to meet the
condition of pH2.9. The condition of the top boundary is obtained by solving the
equilibrium condition of saline with 2.0M CO2 resolved at T=50o and P=100bar. Ca2+ is
added to the boundary inflow like the initial condition. The three reactions and the
concentrations of the top inflow are listed in the Table 13 and Table 15. The flow rate of
the injected water is computed from the average seepage velocity, 𝑣𝑠𝑒𝑒𝑝 = 5.8 ×
10−3 cm/s. The volume flow rate, Q is computed by 𝑄 = 𝜙𝐴𝑣𝑠𝑒𝑒𝑝 = 6.9 × 10−11 m3 /s .
Table 12. Summary of the generated network.
Number of reactive pores
Pore volume
902
6.611×10-4mm3
𝜇𝑉
70
(mm3)
𝜇log𝑉
-3.42
𝜎log𝑉
0.51
Mean of pore surface area
6.003mm2
Surface to volume ratio, S/V
9.081×104cm-1
Porosity
14.01%
Mean of channel conductance, 𝜇Cij
8.086×10-14 m3/Pa∙s
Correlation coefficient, 𝜌 of
0.902
𝑋𝑖𝑗 and 𝑌𝑖𝑗
Mean of channel length
0.000178mm
Mean of channel cross section area
2.295×10-4mm2
Permeability
51mD
Table 13. Three reactions for the condition of the top boundary inflow.
Reactions
logKeq
1.
H2O⇔H++OH-
-13.2(K1)
2.
H2CO3*⇔HCO3-+H+
-6.15(K2)
3.
HCO3-⇔CO32-+H+
-10.0(K3)
Table 14. Initial state of pores. pH is 6.6.
Species
[∙](M) at equilibrium
γ at equilibrium
[∙](M) given
H2O
54.7
-
54.7
71
H+
3.64×10-3
0.686
0.0
OH-
3.64×10-11
0.686
0.0
Ca2+
7.9×10-6
0.221
7.9×10-6
Na+
0.45
-
0.45
Cl-
0.45
-
0.45
Table 15. The condition of the top boundary inflow. The CO2 solubility is 2.0M and pH is 2.9.
Species
[∙](M) at equilibrium
γ at equilibrium
[∙](M) given
H2O
54.7
-
54.7
H+
1.734×10-3
0.686
0.0
H2CO3
1.998
1.0
2.0
HCO3-
1.734×10-3
0.686
0.0
CO32-
4.52×10-10
0.221
0.0
OH-
7.74×10-11
0.686
0.0
Ca2+
7.9×10-6
0.221
7.9×10-6
Na+
0.45
-
0.45
Cl-
0.45
-
0.45
The simulation of CO2 sequestration is performed with the constructed network,
and the result is compared with the published result. The figure shows the mass change
rates and effective reaction rates of anorthite and kaolinite. The effective reaction rate of
anorthite increases from 0 to 6.8×10-9mol/m2s, and the reaction is dissolution. The
effective reaction rate of kaolinite decreases from 0 to -7.7×10-10mol/m2s, and it is
precipitation. The anorthite reaction at the steady state is roughly 10 times as large as that
of kaolinite, and this large reaction rate produces abundant Al3+ and H4SiO4. This large
concentration of Al3+ and H4SiO4 makes the kaolinite reaction in the precipitation
direction. The graphs of the effective reactions of anorthite and kaolinite are very similar
72
but there are small differences between the effective reactions at the steady state. The
effective reactions rates and volume-averaged reaction rates are listed in the Table 16, and
the graphs of reaction rates and various histograms are shown in the Figure 23 and Figure
24. The network for the simulation is randomly generated and the network cannot be
exactly same as that of the published paper. Thus, the error is acceptable, and the
validation of the developed code is completed.
̅ , and the ratio
Table 16. Effective reaction rates, 𝑹𝑵 , volume-averaged reaction rates, 𝑹
̅ /𝑹𝑵 . The result is compared with Li.
𝜷=𝑹
Anorthite(mol/m2s)
Kaolinite(mol/m2s)
𝑅𝑁,𝐴
̅̅̅
𝑅𝐴
𝛽𝐴
𝑅𝑁,𝐾
̅̅̅̅
𝑅𝐾
𝛽𝐾
Simulation result
6.8×10-9
2.4×10-8
3.5
-7.7×10-10
2.4×10-12
-0.0031
Li[4]
~5.5×10-9
~1.2×10-8
2.3
~ -7.5×10-10
~ 1.4×10-12
-0.0018
73
Figure 23. Graphs of the total mass change rates and effective reaction rates.
74
Figure 24. Histograms of pH, [Ca2+], SIA, SIK, and reaction rates at the steady state.
75
b. Application to Viking samples
The simulation of CO2 sequestration is performed with the three Viking samples,
sandstone (Vik3w4), shaly sandstone (Vik10w4), and conglomerate sandstone (Vik14w5).
The real networks of the three rock samples have very different network structures from
the generated regular lattice network in the previous section, and this causes the very
different simulation results from the previous result. The red phase surface area,
(SRG+SRB) and the kaolinite surface area, SGB in the mineral reactivity analysis are used
for the reactive anorthite surface area and the reactive kaolinite surface area respectively.
As shown in the section III.1, the reactive surface area distributions are very different
from the artificial network, and especially, the total reactive surface areas are very
different. In the case of the previous simulation, the reactive anorthite and kaolinite
surface areas are 5% respectively. The Viking samples have approximately 1% of the
reactive anorthite surface and 20~30% of the reactive kaolinite surface area. Besides, the
regular lattice network has only coordination number six of interior pores, while usually,
the graph of the coordination number in the Figure 13 is a typical coordination number
distribution of a rock sample.
The initial state is exactly same as the previous computation. It is the equilibrium
state at T=50oC and P=100bar. The saline water flows from the top boundary to bottom
boundary, and no water flows through side boundaries. The condition of the injected
saline water from the top boundary is different from the previous computation. The
condition is obtained by the solubility computation and the equilibrium computation of
reactions in the table. The CO2 solubility is 1.01M, and the pH is 3.01. The state of the
inflow is shown in the Table 17. In this simulation, the flow rate is a variable which affect
the precipitation of kaolinite. If the flow rate is slow, the acidic saline water stays more
76
time in a reactive pore. The anorthite reaction has more chances to produce Al3+ and
H4SiO4 and to consume H+ turning the kaolinite reaction in the more precipitation
direction. Two seepage velocities, 𝑣seep = 0.0058, 0.001cm/s are selected for the
simulation. The first value 0.0058cm/s is the value from the previous research, and
0.001cm/s is used to see more kaolinite precipitation. For an extreme case, 0.0001cm/s is
used for the simulation with the Viking3W4.
Table 17. The condition of the top boundary inflow. CO2 solubility is 1.01M and pH is 3.01.
Species
[∙](M) at equilibrium
γ at equilibrium
[∙](M) given
H2O
54.7
-
54.7
H+
1.234×10-3
0.686
0.0
H2CO3
1.012
1.0
1.013
HCO3-
1.234×10-3
0.686
0.0
CO32-
4.52×10-10
0.221
0.0
OH-
1.07×10-10
0.686
0.0
Ca2+
7.9×10-6
0.221
7.9×10-6
Na+
0.45
-
0.45
Cl-
0.45
-
0.45
The simulations with Viking samples are performed, and the Figure 25 ~ Figure 37 are
plots and graphs of the result of the simulation. The effective and volume-averaged
reaction rates of different simulations are summarized in the Table 18. The Figure 25,
Figure 26, and Figure 27 show the pH inside the Viking samples at the given times. Red
77
and blue indicate the lowest pH3.0 and the highest pH6.6 respectively. The acidic saline
is injected from the top boundary, and the pH is decreased from pores around the top
boundary. The pH plots show that the network flow simulation works well, and the result
is reasonably good. Next figures show the histograms of pH, [Ca2+], reaction rates, etc,
and the histograms are compared with the histograms of the previous simulation, or the
histograms in the paper[4].
First of all, the simulation time for the steady state is much longer than the
previous simulation. The simulation time of the previous simulation is less than 200
seconds, and the computed solution approaches to the steady state around t=100sec. The
simulation of Viking samples shows apparent changes of the effective reaction rate of
kaolinite after t=200sec, and the simulation time for the steady state is still unknown.
However, it is at least 1000 seconds based on the current graphs of the kaolinite reaction
rate. The simulation time for the steady state mainly depends on the volume flow rate of
the injected saline, and for the smaller seepage velocity, 𝑣seep = 0.001cm/s, it is
difficult to even estimate the time. Because of the limited computation time for the
simulation, the steady state is not obtained yet. The Table 18 shows the effective and
volume-averaged reaction rates at the given times in the table, and so, the reaction rates
can be different from the values at the steady state. As shown in the Table 18, the ratio,
𝛽𝐴 of anorthite reaction rates are close to one, i.e., the volume-averaged reaction rates are
enough to estimate the overall anorthite reaction rates over the Viking samples. The ratio,
𝛽𝐾 of kaolinite reaction rates are also close to one except in the case of Viking3W4 with
𝑣seep = 0.001cm/s. The difference between the true effective reaction rate and volumeaveraged reaction rate of kaolinite is caused by the interaction of kaolinite dissolution and
precipitation. If there is only one type of reactions, dissolution or precipitation, the
78
volume-averaged reaction rate is not far from the effective reaction rate. The anorthite
reaction is always dissolution, and the anorthite precipitation never appears in the whole
simulation of Viking samples. Thus, the ratio, 𝛽𝐴 is around 1. There appear a few pores
where the kaolinite reaction is precipitation around t=100sec, and later, more pores of
kaolinite precipitation appear. These pores of kaolinite precipitation start to change the
volume-averaged reaction rates from the effective reaction rates. In all simulation cases,
the kaolinite precipitation increases as time goes. Moreover, the ratio, 𝛽𝐾 of Viking3W4
with 𝑣seep = 0.001cm/s can be larger than 2.9 at a later time. If the effective reaction
rate of kaolinite decreases to zero, the ratio, 𝛽𝐾 is extremely large. In the simulation
with very slow seepage velocity, 𝑣seep = 0.0001cm/s, the all kaolinite reaction is
precipitation, and thus, the ratio, 𝛽𝐾 is stably close to one. The ratio, 𝛽𝐾 of Viking14w5
is very close to one, but this value is obtained at an early time when the heterogeneity
effect by the interaction of kaolinite dissolution and precipitation is not developed yet.
The histograms in the Figure 31~Figure 37 are very different from the histograms
of the previous simulation with the artificial uniform lattice network. The ranges and
distributions of pH, saturation index (SI), and reaction rates are very different from the
histograms in the previous section. The difference in the histograms is expected to be
mainly caused by the different network. The histograms are different partially because the
simulations are computed to the steady state. There are small difference between the state
of the injected saline, and this can cause some difference. The difference should be
carefully analyzed in the geochemical aspect to obtain a practical interpretation of the
simulation of the Viking samples.
79
Table 18. Simulation result of the effective and volume-averaged reaction rates. The
reaction rates are computed at the times given in the table.
Anorthite(mol/m2s)
𝑣seep (cm/s)
0.0058
Kaolinite(mol/m2s)
𝑅𝑁,𝐴
̅𝑅̅̅𝐴̅
𝛽𝐴
𝑅𝑁,𝐾
̅̅̅̅
𝑅𝐾
𝛽𝐾
1.51×10-8
1.58×10-8
1.05
1.83×10-12
2.15×10-12
1.17
9.63×10-9
1.00×10-8
1.04
5.03×10-13
1.46×10-12
2.90
3.14×10-9
3.23×10-9
1.03
-9.69×10-12
-8.76×10-12
0.90
1.67×10-8
1.79×10-8
1.07
2.21×10-12
2.24×10-12
1.01
1.22×10-8
1.34×10-8
1.10
1.99×10-12
2.07×10-12
1.04
(t=810sec)
Viking
0.001
3W4
(t=1227sec)
0.0001
(t=1386)
0.0058
Viking
(t=)
10W4
0.001
(t=)
0.0058
Viking
(t=350)
14W5
0.001
(t=400)
80
pH 3.0
6.6
(a) t=0
(b) t=0.1sec
(c) t=1sec
(d) t=10sec
Figure 25. pH plots in the primary pore space of Viking3W4. Grey and dark grey represent
quartz and MoI respectively. The CO2 resolved saline is injected from top to bottom with
𝒗𝒔𝒆𝒆𝒑 = 𝟎. 𝟎𝟎𝟓𝟖𝐜𝐦/𝐬.
81
pH 3.0
6.6
(a) t=0.0
(b) t=1.7sec
(c) t=6.0sec
(d) t=91.0sec
Figure 26. pH plots in the primary pore space of Viking10W4. Grey and dark grey
represent quartz and MoI respectively. The CO2 resolved saline is injected from top to
bottom with 𝒗𝒔𝒆𝒆𝒑 = 𝟎. 𝟎𝟎𝟓𝟖𝐜𝐦/𝐬.
82
pH 3.0
6.6
(a) t=0
(b) t=3.0sec
(c) t=10.0sec
(d) t=450.0sec
Figure 27. pH plots in the primary pore space of Viking14W5. Grey and dark grey
represent quartz and MoI respectively. The CO2 resolved saline is injected from top to
bottom with 𝒗𝒔𝒆𝒆𝒑 = 𝟎. 𝟎𝟎𝟓𝟖𝐜𝐦/𝐬.
83
Figure 28. Mass change rates and effective reaction rates of anorthite and kaolinite.
Viking3W4. 𝒗𝒔𝒆𝒆𝒑 = 𝟎. 𝟎𝟎𝟓𝟖𝐜𝐦/𝐬. The dotted lines in the graphs of the effective reaction
rates indicate the volume-averaged reaction rates.
84
Figure 29.
Mass change rates and effective reaction rates of anorthite and kaolinite.
Viking3W4. 𝒗𝒔𝒆𝒆𝒑 = 𝟎. 𝟎𝟎𝟏𝐜𝐦/𝐬. The dotted lines in the graphs of the effective reaction
rates indicate the volume-averaged reaction rates.
85
Figure 30. Mass change rates and effective reaction rates of anorthite and kaolinite.
Viking3W4. 𝒗𝒔𝒆𝒆𝒑 = 𝟎. 𝟎𝟎𝟎𝟏𝐜𝐦/𝐬. The dotted lines in the graphs of the effective reaction
rates indicate the volume-averaged reaction rates.
86
Figure 31. Histograms of pH, [Ca2+], SI’s, and reaction rates at t=810sec. Viking3W4.
𝒗𝒔𝒆𝒆𝒑 = 𝟎. 𝟎𝟎𝟓𝟖𝐜𝐦/𝐬.
87
Figure 32. Histograms of pH, [Ca2+], SI’s, and reaction rates at t=1223sec. Viking3W4.
𝒗𝒔𝒆𝒆𝒑 = 𝟎. 𝟎𝟎𝟏𝐜𝐦/𝐬.
88
Figure 33. Histograms of pH, [Ca2+], SI’s, and reaction rates at t=1223sec. Viking3W4.
𝒗𝒔𝒆𝒆𝒑 = 𝟎. 𝟎𝟎𝟎𝟏𝐜𝐦/𝐬.
89
Figure 34. Mass change rates and effective reaction rates of anorthite and kaolinite.
Viking14W5. 𝒗𝒔𝒆𝒆𝒑 = 𝟎. 𝟎𝟎𝟓𝟖𝐜𝐦/𝐬. The dotted lines in the graphs of the effective reaction
rates indicate the volume-averaged reaction rates.
90
Figure 35.
Mass change rates and effective reaction rates of anorthite and kaolinite.
Viking14W5. 𝒗𝒔𝒆𝒆𝒑 = 𝟎. 𝟎𝟎𝟏𝐜𝐦/𝐬. The dotted lines in the graphs of the effective reaction
rates indicate the volume-averaged reaction rates.
91
Figure 36. Histograms of pH, [Ca2+], SI’s, and reaction rates at t=1223sec. Viking14W5.
𝒗𝒔𝒆𝒆𝒑 = 𝟎. 𝟎𝟎𝟓𝟖𝐜𝐦/𝐬.
92
Figure 37. Histograms of pH, [Ca2+], SI’s, and reaction rates at t=1223sec. Viking14W5.
𝒗𝒔𝒆𝒆𝒑 = 𝟎. 𝟎𝟎𝟏𝐜𝐦/𝐬.
93
IV. Conclusion & future study
We develop a full process for the network flow simulation of CO2 sequestration
with X-ray CT images of rock samples, and we apply the simulation method to three
Viking samples. The simulation process consists of a few steps; mineral identification,
construction of pore-throat network of mineral distributions, single phase network flow
model for the advective transport, and network flow simulation of CO2 sequestration with
reaction models. Each step is well established and proved. The mineral identification and
pore-throat network of mineral distributions are successfully completed, and the obtained
result is reasonably acceptable. The result is comparable with the BSE analysis. The
single-phase network flow model is also developed, applied, and proved. The model is
compared with the experimental model, and it agrees well with the experimental model.
The reaction models of the network flow model for CO2 sequestration are proved by
applying the reaction models to the previous result. The whole network flow model for
CO2 sequestration is applied to Viking samples to compute the effective reaction rates
and capture the reaction heterogeneities which are impossible to see with larger scale
simulation. The behaviors of the reaction rates are very different from the published
result, and it also depends on the volume flow rate of the injected saline. The pore-pore
heterogeneity is varied by the volume flow rate too. The time scale for the simulation is
order of 1000 seconds while that of the previous paper is order of 100 seconds. The large
time scale results in the large computation time, and thus, a new scheme should be added
for fast simulation. We can use the result for further analyses such as macro scale
94
simulation of CO2 sequestration, estimation the capacity of CO2 storage, etc.
However, there are a few future topics required for more accurate, efficient
simulation. First, the dissolution and precipitation change the porous structure. By the
two kinetic reactions, there are dissolution and precipitation in the core sample, and the
dissolution and precipitation change the geometry of the porous media. The network
structure is transformed by this geometrical change, and the transport property of the
reactive flow can be changed. The geometrical effect of dissolution and precipitation
should be considered for a more accurate simulation. Second, the mineral of red phase
covered by kaolinite is partially accessible. The kaolinite is permeable, but it is less
permeable than void phase. In the full temporal scale of CO2 sequestration, the MoI
covered by kaolinite is completely accessible, but in the shorter temporal scale, it may not
be accessible. A model for the partial permeable property of the kaolinite to the MoI is
another future work. Third, the computation time of the simulation increases
proportionally to the number of pores, and the simulation of Viking3W4 which has the
most pores requires so much computation time. To reduce the computation time, a new
scheme such as implicit time integration is needed to make time step larger. The network
flow model has the reaction source term, which makes the implicit scheme complicated
and less effective. Another approach for fast simulation is to parallelize the code. Finally,
additional reaction models with minerals other than anorthite and kaolinite are another
key to improve the simulation result. The phase of minerals of interest has many other
minerals, and so, there are possibly some other important kinetic reactions which should
be included in the network flow simulation. There is rarely an intensive study about the
reaction models in the CO2 sequestration, and we need a careful inspection on the
reaction models in the CO2 sequestration. Many experiments is required to build a new
95
reaction model.
96
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