INFLUENCE OF CAPILLARY PRESSURE
ON CO2 STORAGE AND MONITORING
Juan E. Santos
Purdue University
and Instituto del Gas y del Petróleo de la Univ. de Buenos Aires (IGPUBA),
Argentina and Univ. Nac. de La Plata
Work in collaboration with: G. B. Savioli (IGPUBA), L. A. Macias (IGPUBA),
J. M. Carcione and D.Gei ((OGS) Trieste, Italy)
Introduction. I
 CO2 sequestration in suitable geological formations is one of the
solutions to mitigate the greenhouse effect.
 Saline aquifers are suitable as storage sites due to their large volume
and their common occurrence in nature.
 Numerical modeling of CO2 injection and seismic monitoring are
important tools to understand the long term behavior after injection
and to test the effectiveness of CO2 sequestration.
Introduction. II
The first industrial CO2 injection project started in 1996 is at the
Sleipner gas field in Norway.
The CO2 separated from
natural gas is being
injected in a saline
aquifer, the
Utsira formation
a high permeable
sandstone with several
mudstone layers that
limit the vertical motion
of the CO2
From: http://decarboni.se/publications
Introduction. III
 We introduce a numerical procedure combining simulations of:
• CO2 injection and storage in saline aquifers.
• Seismic monitoring of CO2 migration in the subsurface.
 The multiphase flow functions (relative permeability and capillary
pressure relations) are determined from on-site resistivity
measurements.
 In particular we analyze the sensitivity of the spatial distribution
of CO2 and their seismic images due to capillary pressure
variations.
Methodology
1. Black-Oil simulator to model CO2 injection and storage.
• Multiphase Flow Functions
• A model to update the petrophysical properties
2. Seismic monitoring using a viscoelastic model formulated in the
space-frequency domain that
attenuation and dispersion effects.
includes
mesoscopic-scale
The Black-Oil formulation
Mass conservation equation
1
𝑅𝑠
−𝛻 ∙
𝑣 + 𝑣 + 𝑞𝑔 =
𝐵𝑔 𝑔 𝐵𝑏 𝑏
𝜕 𝜙
1
−𝛻 ∙
𝑣 + 𝑞𝑏 =
𝐵𝑏 𝑏
𝑏
brine
𝑔
CO2
𝜙: porosity
𝑆𝑖 : phase 𝑖 saturation
𝑝𝑖 : phase 𝑖 pressure
𝑞𝑖 : flow rate per unit volume
𝑆𝑔 𝑅𝑠 𝑆𝑏
+
𝐵𝑔
𝐵𝑏
𝜕𝑡
𝑆𝑏
𝐵𝑏
𝜕𝑡
𝜕 𝜙
Darcy’s Empirical Law
𝑘𝑟𝑔
𝑣𝑔 = −𝐾
𝛻𝑝𝑔 − 𝜌𝑔 𝑔𝛻𝑧
𝜇𝑔
𝑘𝑟𝑏
𝑣𝑏 = −𝐾
𝛻𝑝𝑏 − 𝜌𝑏 𝑔𝛻𝑧
𝜇𝑏
𝑆𝑏 + 𝑆𝑔 = 1
𝑝𝑔 − 𝑝𝑏 = 𝑃𝐶 𝑆𝑏
𝜇𝑖 : phase 𝑖 viscosity
𝑅𝑠 : 𝐶𝑂2 solubility in brine
𝐵𝑔 : 𝐶𝑂2 formation volume factor 𝐾: absolute permeability tensor
𝐵𝑏 : brine formation volume factor 𝑘𝑟𝑖 (𝑆𝑖 ): phase 𝑖 relative permeability
𝑃𝐶 𝑆𝑖 : capillary pressure
The Black-Oil formulation - BOAST
• The numerical solution was obtained employing the public domain
software BOAST.
• BOAST solves the flow differential equations using IMPES (IMplicit
Pressure Explicit Saturation), a finite difference technique.
• The basic idea of IMPES is to obtain a single equation for the brine
pressure by a combination of the flow equations. The system is
linearized evaluating kr and PC at the saturations of the previous time
step. Once pressure is implicitly computed for the new time step,
saturation is updated explicitly.
Multiphase Flow Functions – Resistivity Index
The multiphase flow functions were obtained from the Resistivity Index (RI)
Using the log data and the conductivity relation (Carcione et. al. JPSE, 2012):
𝜎 𝑆𝑏 = 1 − 𝜙
At 𝑆𝑏 = 1
𝛾
𝜎𝑞
+𝜙 1−
𝛾
𝑆𝑏 𝜎𝑔
+
𝛾
𝛾
𝜎 1 = 1 − 𝜙 𝜎𝑞 + 𝜙𝜎𝑏
𝛾: 𝑓𝑟𝑒𝑒 𝑝𝑎𝑟𝑎𝑚𝑒𝑡𝑒𝑟 (0.75)
𝜎𝑏 : 𝑏𝑟𝑖𝑛𝑒 𝑐𝑜𝑛𝑑𝑢𝑐𝑡𝑖𝑣𝑖𝑡𝑦
𝜎𝑞 : 𝑔𝑟𝑎𝑖𝑛 𝑐𝑜𝑛𝑑𝑢𝑐𝑡𝑖𝑣𝑖𝑡𝑦
𝑅𝐼 𝑆𝑏 : 𝑟𝑒𝑠𝑖𝑠𝑡𝑖𝑣𝑖𝑡𝑦 𝑖𝑛𝑑𝑒𝑥
from logs
at Utsira
Determines 𝜎𝑞
𝜎 𝑆𝑏 : 𝑐𝑜𝑛𝑑𝑢𝑐𝑡𝑖𝑣𝑖𝑡𝑦
𝜎𝑔 : 𝐶𝑂2 𝑐𝑜𝑛𝑑𝑢𝑐𝑡𝑖𝑣𝑖𝑡𝑦
𝛾
𝜙𝑆𝑏 𝜎𝑏
Then:
𝑅𝐼 𝑆𝑏
𝜎 1
=
𝜎 𝑆𝑏
Multiphase Flow Functions – Relative Permeability Curves
Relative permeability curves are obtained from RI(Sb):
𝑘𝑟𝑔 𝑆𝑏 = 1 − 𝑆𝑏∗
1
Brine
1
2
1 − 𝑘𝑟𝑏 𝑆𝑏
Relative Permeability
𝑘𝑟𝑏 𝑆𝑏
𝑆𝑏∗
=
𝑅𝐼 𝑆𝑏
CO2
1
1
0
0
𝑆𝑏 − 𝑆𝑟𝑏
∗
𝑆𝑏 =
1 − 𝑆𝑟𝑏
𝑆𝑟𝑏 : 𝑟𝑒𝑠𝑖𝑑𝑢𝑎𝑙 𝑏𝑟𝑖𝑛𝑒 𝑠𝑎𝑡𝑢𝑟𝑎𝑡𝑖𝑜𝑛
0
0,1
0,2
0,3
0,4
0,5
Brine Saturation
0,6
0,7
0,8
Multiphase Flow Functions – Capillary Pressure Curve
The Pc function is represented using an exponential model with a free
parameter 𝑛 obtained adjusting the brine permeability curve:
𝑃𝑐𝐷 𝑆𝑏 = 𝑆𝑏∗
−1 𝑛
2+𝑛
𝑛
2,50
𝑛
𝑃𝑐 𝑆𝑏 = 𝑃𝑐𝐷 𝑃𝑐𝑒
𝑃𝑐𝑒 : 𝑡ℎ𝑟𝑒𝑠ℎ𝑜𝑙𝑑 (𝑒𝑛𝑡𝑟𝑦) 𝑐𝑎𝑝𝑖𝑙𝑙𝑎𝑟𝑦 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒
Capillary Pressure (kPa)
𝑘𝑟𝑏 𝑆𝑏
𝑆𝑏∗
=
= 𝑆𝑏∗
𝑅𝐼 𝑆𝑏
Pce=1kPa
2,00
1,50
1,00
0,50
0,00
0
0,1
0,2
0,3
0,4
0,5
CO2 Saturation
𝑃𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝑎𝑡 𝑤ℎ𝑖𝑐ℎ 𝑡ℎ𝑒 𝑔𝑎𝑠 𝑝ℎ𝑎𝑠𝑒 𝑖𝑠 𝑠𝑢𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡𝑙𝑦 𝑐𝑜𝑛𝑛𝑒𝑐𝑡𝑒𝑑 𝑡𝑜
𝑎𝑙𝑙𝑜𝑤 𝑓𝑙𝑜𝑤.
𝑷𝒄𝒆 𝒘𝒊𝒍𝒍 𝒃𝒆 𝒖𝒔𝒆𝒅 𝒊𝒏 𝒐𝒖𝒓 𝒔𝒆𝒏𝒔𝒊𝒕𝒊𝒗𝒊𝒕𝒚 𝒂𝒏𝒂𝒍𝒚𝒔𝒊𝒔.
0,6
0,7
0,8
A Model to update the Petrophysical properties
Carcione’s model (Carcione et.al., IJRMMS, 2003)
𝑝 𝑡 = 𝑆𝑏 𝑝𝑏 𝑡 + 𝑆𝑔 𝑝𝑔 𝑡
1 − 𝜙𝑐
𝜙 𝑡
𝑝 𝑡 − 𝑝𝐻 = 𝜙0 − 𝜙 𝑡 + 𝜙𝑐 ln
𝐾𝑠
𝜙0
45 1 − 𝜙 𝑡
1
=
𝑘𝑥 𝑡
𝜙 𝑡 3
2
1−𝐶
𝑅𝑞2
2
𝐶2
+ 2
𝑅𝑐
𝑘𝑥 𝑡
1 − 1 − 0.3𝑎 sin 𝜋𝑆𝑏
=
𝑘𝑧 𝑡
𝑎 1 − 0.5 sin 𝜋𝑆𝑏
𝑝: pore pressure
𝐶: clay content
𝜙𝑐 : critical porosity
𝜙0 : initial porosity
𝑅𝑞 : radius of sand grains
𝑅𝑞 : radius of clay grains
𝑝𝐻 : hydrostatic pore pressure
𝐾𝑠 : bulk modulus of solid grains
𝑎: permeability anisotropy parameter
Seismic Modeling – Mesoscopic Attenuation Effects
• Within the Utsira formation and outside the mudstone layers, we
determine the complex and frequency dependent P-wave modulus
𝐸 𝜔 = 𝜆 𝜔 + 2𝜇 𝜔
at the mesoscale using White’s theory for patchy saturation.
𝜆 𝜔 , 𝜇 𝜔 : Complex Lamé coefficients
𝜔: angular frequency
Seismic Modeling – Constitutive Relations
𝑢 = 𝑢 𝜔 = 𝑢𝑥 𝜔 , 𝑢𝑧 𝜔
𝜎𝑗𝑘 𝑢 = 𝜆 𝜔 𝑑𝑖𝑣 𝑢 𝛿𝑗𝑘 + 2𝜇 𝜔 𝜀𝑗𝑘 𝑢
𝑢: time-Fourier transform of the displacement vector
𝜎𝑗𝑘 𝑢 : stress tensor
𝜀𝑗𝑘 𝑢 : strain tensor
𝛿𝑗𝑘 𝑢 : Kroenecker delta
Seismic Modeling – Phase Velocities and Attenuation coefficients
1
𝑣𝑖 𝜔 = 𝑅𝑒
𝑣𝑐𝑖
𝑣𝑐𝑝 𝜔 =
−1
,
𝑄𝑖 𝜔 =
𝐸 𝜔
𝜌
𝑅𝑒 𝑣𝑐𝑖 𝜔 2
𝐼𝑚 𝑣𝑐𝑖 𝜔 2
𝑣𝑐𝑠 𝜔 =
,
𝜇 𝜔
𝜌
𝑣𝑖 𝜔 : frequency dependent phase velocities
𝑄𝑖 𝜔 : quality factors
𝜌: bulk density
𝑣𝑐𝑝 𝜔 : compressional velocity
𝑣𝑐𝑠 𝜔 : shear velocity
𝑖 = 𝑝, 𝑠
Seismic Modeling – A Viscoelastic Model for Wave Propagation
Equation of motion in a 2D viscoelastic domain Ω:
𝜔2 𝜌𝑢 + 𝛻. 𝜎 𝑢 = 𝑓 𝑥, 𝜔 ,
−𝜎 𝑢 𝜈 = 𝑖𝜔𝐷𝑢,
𝑓 𝑥, 𝜔 : external source
Ω
𝜕Ω
𝐷: positive definite matrix
𝜈 = 𝜈1 , 𝜈2 : unit outward normal
𝜈1
𝐷 = 𝜌 −𝜈
2
𝜈2 𝑣𝑝 𝜔
𝜈1
0
0
𝑣𝑠 𝜔
𝜈1
𝜈2
−𝜈2
𝜈1
Finite Element Method
• The FE space-frequency solution of the viscoelastic wave equation was
computed at a selected number of frequencies in the range of interest using an
iterative FE domain decomposition procedure.
• To approximate each component of the solid displacement vector we employed
a nonconforming FE space 𝑁𝐶 ℎ defined over a partition of the domain Ω into
rectangles of diameter bounded by ℎ.
• The use of the FE space 𝑁𝐶 ℎ generates less numerical dispersion than the
standard bilinear elements. The error of the FE procedure is of order ℎ1/2
(Santos et. al., CMAME, 2002 ).
• The time domain solution was obtained using a discrete inverse Fourier
transform.
Aquifer Model
Utsira formation
0,4 km depth
2,5D model
1,2 km length
10 km thickness
Fractal Initial Porosity
Fractal Initial Permeability
Low Permeability Mudstones
From: http://www.sintef.no
Aquifer Model – Initial Vertical Permeability
Within the formation there are several mudstone layers which act as
barriers to the vertical motion of the CO2.
Initial Vertical
Permeability
Distribution
[mD]
Sensitivity analysis - saturation maps after 3 years of injection
Pce=5kPa
Pce=200kPa
Sensitivity analysis – Qp and vp after 3 years of injection
Pce=5kPa
Pce=200kPa
Qp
vp
Sensitivity analysis – synthetic seismograms after 3 years of injection
Pce=5kPa
300
1200
0
Time (s)
Time (s)
0
Distance (m)
600
900
Pce=200kPa
300
Distance (m)
600
900
1200
Saturation maps up to 7 years of injection
Pce=200kPa
Synthetic seismogram after 7 years of injection
0
300
Distance (m)
600
900
1200
Pce=200kPa
Time (s)
Real seismogram
50ms
50
Chadwick et. al., BGS, (2004)
Time-lag (ms)
0
Conclusions
• The fluid-flow simulator yields CO2 accumulations below the
mudstone layers and the corresponding synthetic seismograms
resemble the real data, where the pushdown effect is clearly
observed.
• Capillary forces affect the migration and dispersal of the CO2 plume;
higher values of the threshold capillary pressure Pce cause slower
CO2 upward migration and thicker zones of CO2 accumulations.
• Variations in capillary forces induce noticeable changes in the seismic
images of the Utsira formation, clearly seen in the synthetic
seismograms.
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