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Stanford UniversityGlobal Climate & Energy Project
Numerical Simulation Framework for CO2 Sequestration
Hamdi Tchelepi, Lou Durlofsky, Khalid Aziz
Department of Energy Resources EngineeringStanford University
GCEP SymposiumSeptember 20, 2006
2
Different Sources, Varying Quality & QuantityMulti-Scales, Multi-Physics
Upscaling Downscaling
10-5 10-4 10-3 10-2
10-1 100 101 102 103 104 105 106 107 108 109 1010
Thin Sections
Core Data
Well Log
Geological Model Cells
Up-scaled Simulation
Cells
Well Test
Seismic Data
Heterogeneous, Large Systems Sparse Data
3
CO2 SequestrationModeling Framework
• Engineering Tool: Design & Management of CO2 Sequestration Projects– Advanced & smart wells – Control & optimization– Long-term monitoring
• Physics and Numerics– Detailed study and modeling of the physics– Robust, efficient numerical algorithms
• Flexible Extensible Computational Framework– Incorporate research results
4
Fundamental physics
Multiscale
Generalized Compositional
Formulation
Adaptive Implicit
General PurposeResearch Simulator
Opt
imiz
atio
n
CO2 SequestrationComputational Framework
Advanced Wells
5
Research Activity
• Basic GPRS Capabilities - Fan, Pan
• Miscible & Immiscible Plumes - Riaz, Hesse
• Gravity Currents - Hesse with Lynn
• MultiScale Formulation - Zhou
• High-order AIM - De Louben, Riaz
• Particles for Nonlinear Flow – Tyagi with Prof. Jenny of ETH
6
GPRS Extensions
• Residual Trapping – Relative Permeability Hesteresis
– Preliminary Results: Injection Strategies
• Diffusion and Dispersion
• Fast Flash for CO2-Water Systems
7
SJ Saline Aquifer Modeling
• Size: 260,000 x 98,000 x 980 ft3
• Depth: 2000 ft• Porosity: 0.135• Temperature: 104o F• Grid:160 x 60 x 20• Permeability:
• Aquifer data determined from existing hydrological data (BEG database, UT Austin)
Generated by S-GeMS
8
Well Configuration and BCs
• One CO2 injection well completed in bottom layer
• CO2 rate 141,300 MCF/day (29M t/year)
INJ
Constant pressure
(continuous aquifer)
No-flow BC
9
Simulation Results
Sg at 100 years (80 years of well shut-in) without hysteresis
Sg after CO2 injected for 20 years
10 20 30 40 50 60
5
10
15
20
0.2
0.4
0.6
0.8
10 20 30 40 50 60
5
10
15
20
0.2
0.4
0.6
0.8
Sg at 100 years (80 years of well shut-in) with hysteresis
10 20 30 40 50 60
5
10
15
20
0.2
0.4
0.6
0.8
10
Research Activity
• Basic GPRS Capabilities - Fan, Pan
• Miscible & Immiscible Plumes - Riaz, Hesse
• Gravity Currents - Hesse with Lynn
• MultiScale Formulation - Zhou
• High-order AIM - De Louben, Riaz
• Particles for Nonlinear Flow – Tyagi with Prof. Jenny of ETH
12
Why is Convection Important?
Dissolution of CO2 increases significantly!
Hesse, Riaz, Tchelepi, Orr
13
Summary
• Convection can be important: Constant dissolution rate⇒ Dissolved CO2 increases linearly
• Effective in large, high K aquifers: Onset time & critical wavelength decrease with K
• Not resolving the instability shifts scales
• Heterogeneity, anisotropy
102
101
100
10-1
102 10410310-2
diffusive
~diffusiveconvective
Ra
t
17
M =1/50, G = 20
ViscouslyUnstable
ViscouslyStable
Brine
CO2
μμM =
Large Differences inDensity & Viscosity
18
Immiscible Plumes: Remarks
• Linear analysis & high resolution simulations of immiscible two-phase flow (injection & post-injection)
• Complex behaviors in the presence of density and viscosity differences
• Post-injection modeling challenges– Unstable drainage– Residual trapping
19
Research Activity
• Basic GPRS Capabilities - Fan, Pan
• Miscible & Immiscible Plumes – Riaz, Hesse
• Gravity Currents - Hesse with Lynn
• MultiScale Formulation - Zhou
• High-order AIM - De Louben, Riaz
• Particles for Nonlinear Flow – Tyagi with Prof. Jenny of ETH
21
Simple Analytical Model
Consider two fluids separated by a sharp interface, the equation for the evolution of the height h(x,t) interface is:
⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
+−−
∂∂
=∂∂
xh
HMhhHh
xth
)1()(κ
g
rgkkgφμρ
κ*Δ
=grw
wrg
w
g
kk
Mμμ
λλ
*
*
==
23
Many mid-continental saline aquifers are gently sloping, and lack a structural trap.
How does the maximum migration distance change with increasing slope?
Storage in Open Sloping Aquifers
24
Simple Model of Residual Trapping(preliminary results!)
Incorporate loss of a constant residual saturation into gravity current model:
The effect of small slope on residual trapping?
⎟⎠⎞
⎜⎝⎛
∂∂
∂∂
=∂∂
+∂∂
xhh
xtx
xhtx
th θκθκ cos),(sin),(
( )
( )⎪⎪⎩
⎪⎪⎨
⎧
>∂∂
−Δ
=
≤∂∂
−−Δ
==
0;1
0;1),(
1
1
th
Skg
th
SSkg
tx
wr
g
grwr
g
φρλ
κ
φρλ
κκ
27
Gravity Currents: Remarks
• Scaling law of gravity-current tip changes when plume stops feeling aquifer thickness
• Residual trapping increases dramatically with aquifer slope
• Two regimes for a sloping aquifer– Initial power law decay (slumping > sliding)– Late stage with rapid decay (sliding > slumping)
• Migration distance decreases as slope increases• New family of similarity solutions• Improved trapping model!
28
Research Activity
• Basic GPRS Capabilities - Fan, Pan
• Miscible Convection - Riaz, Hesse
• Immiscible Plume Migration - Riaz
• Gravity Currents - Hesse, with Lynn
• MultiScale Formulation - Zhou
• High-order AIM - De Louben, Riaz
• Particles for Nonlinear Flow -Tyagi, Jenny at ETH
29
Motivation
• Interested in modeling flow in large-scale, highly heterogeneous formations
• Multiscale Formulations– Construct & solve coarse-scale problem– Reconstruct fine-scale solution locally– Existing methods deal with incompressible flow
• Objective– Multiscale method for compressible
multiphase flow– Algebraic framework
30
Operator Based Multiscale Method
• Fine scale system
• Construct interpolation & projection operators
• Construct coarse scale system
• Reconstruction of fine-scale pressure
f f f=A p r
[ ] c f c c c= ⇒ =RAP p Rr A p r
f c=p Pp
,P R
31
Operator Based MSFV
• Compressible flow equation
– Existing multiscale methods for elliptic problems– General pressure equation is parabolic
• OBMM( ) [ ]
( )( ) ( ) [ ]
A
A
f A
c c c c
f A
p dV
c x p x dV
λΩ
Ω
⎫∇ ⋅ ∇ =⎪⇒ − =⎬⎪=⎭
∫
∫
c
c
RT PpT C p r
RC Pp
( ) ( )0t fw w fo o Rpp C S C S C qt
λ φ ∂∇ ⋅ ⋅ ∇ = + + +
∂k
32
Prolongation Operator
• The basis functions are given by
• Assemble prolongation operator
I-1 I I+1
i-4 i-3 i-2 i-1 i i+1 i+2 i+3 i+4
D-1 D
( )
1
,[ ] ( )
cnI
A AI
a A A ax
φ φ
φ=
=
=
∑P
“A” denotes coarse node;
“a” denotes fine node
A
ΙΩ%
( )( )( )
( ) ( )
0 in
0 on
IA I
IA
IIt j t
IA B AB
x x
x
λ φ
φλ
φ δ
∇ ⋅ ∇ = Ω
⎛ ⎞∂ ∂= ∂Ω⎜ ⎟∂ ∂⎝ ⎠
=
%
%
33
Restriction operator for MSFV
• FVM equations in fine and coarse scale
• The Restriction operator sums the fine scale equations to form the coarse scale formula
( ) ( ) ( )
( ) ( ) ( )
d d 1,...,
d d 1,...,
a a
A A
f
c
pp V c x V a ntpp V c x V A nt
λ
λ
Ω Ω
Ω Ω
∂∇ ⋅ ∇ = =
∂∂
∇ ⋅ ∇ = =∂
∫ ∫
∫ ∫
[ ] ,
1 if ( 1,..., ; 1,..., )
0 otherwisea A
c fA aR A n a n
⎧ Ω ⊂ Ω= = =⎨⎩
34
Compressible Two-Phase System
• Depletion of liquid-gas reservoir– Initially 50% liquid and 50% gas– The PVT properties for the two fluids are
– Compressibility driven flow– SPE 10 top layer (220 X 60)
30
0
1 10 /1 /
l
g
b p pb p p
−= += +
0
0
0
0
0
0
−4
−2
0
2
4
6
8
35
Algebraic MSFV: Pressure Field
Fine220 x 60
Multiscale22 x 6
50 100 150 200
0
0
0
0
0
0
50 100 150 200
0
0
0
0
0
0
50 100 150 20
0
0
0
0
0
0
50 100 150 200
0
0
0
0
0
0
0.5t τ=
0.005t τ=
0.05t τ=
36
Operator Based Multiscale Method
• Algebraic multiscale framework for high resolution modeling of CO2 Sequestration
• Advantages of OBMM– Extendible to unstructured grid– Easier to include more complicated physics– Allows for incorporating a multiscale formulation into
existing reservoir simulators
• Adaptivity, GPRS implementation• Multiscale formulation for nonlinear transport
37
Research Activity
• Basic GPRS Capabilities - Fan, Pan
• Miscible Convection - Riaz, Hesse
• Immiscible Plume Migration - Riaz
• Gravity Currents - Hesse, with Lynn
• MultiScale Formulation - Zhou
• High-order AIM - De Louben, Riaz
• Particles for Nonlinear Flow -Tyagi, Jenny at ETH
38
CO2 SequestrationMultiscale Multi-Physics
Pore Scale
• Capillary Forces
• Stokes Flow
• Pore Network Simulation
• Statistical Theories
• Invasion Percolation
• DLA, Anti-DLA
Darcy Scale
• Viscous & Gravity Forces
• Darcy’s Law
• Transport:
Eulerian Deterministic
Relative Permeability
StatisticalInformation
Small Scale Large Scale
Stochastic Model
39
Phase transport equation
Conservation of total mass
Elliptic equation for pressure
ααα
qvt
S=⋅∇+
∂∂
qqvv =∑=⋅∇=∑⋅∇ αα
qpkkr −=⎟⎟⎠
⎞⎜⎜⎝
⎛∇⋅∇ ∑ α
α
φ
Transport
Flow
Stochastic Model: Lagrangian Framework• Particle based method (Monte Carlo)• A particle represents a phase (physical particles)• Different from characteristic methods• Particle evolution: statistical rules (pore-scale physics)• Natural modeling of multiscale, multi-physics processes
Modeling Framework
40
2D Test Case
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1 1.2dia length (m)
S
Particle MethodFVM_50_50FVM_100_100
Number of particles per cell 800PVI=0.5
Saturation curve of injected phase along diagonal after 0.5 PVI
42
2D, Two-Phase, Homogeneous
PVI=0.5 PVI=0.7
Before Breakthrough After Breakthrough
Particle Distribution
44
Particle Method: Remarks
• Developed a stochastic framework that provides a consistent link between small and large scales
• Showed how stochastic particles can be used to solve nonlinear conservation equations
• Validation against exact solutions and FVM
• Demonstrate power of the method:– Statistical information from pore scale physics– Particle velocity pdf & multi-point statistics– Non-equilibrium: hysteresis, trapping, reactions– Pore-scale instability, …
45
Fundamental physics
Multiscale
Generalized Compositional
Formulation
Adaptive Implicit
General PurposeResearch Simulator
Opt
imiz
atio
n
CO2 SequestrationComputational Framework
Advanced Wells
46
Research Activity
• Basic GPRS Capabilities - Fan, Pan
• Miscible & Immiscible Plumes – Riaz, Hesse
• Gravity Currents - Hesse with Lynn
• MultiScale Formulation - Zhou
• High-order AIM - De Louben, Riaz
• Particles for Nonlinear Flow – Tyagi with Prof. Jenny of ETH
47
Next Steps
• Continue investigation of post-injection miscible & immiscible CO2-water systems
• Stochastic particle method for linking small and large scales
• Further develop and implement new algorithms in GPRS (e.g., OBMM, High-Order AIM)
• GPRS-based optimization of CO2injection strategies using advanced wells
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