1

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

11

t=0.98

t=1.80

t=2.25

Miscible Convection

Ra = 4000

Riaz, Hesse, Tchelepi, Orr

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

14

Miscible

Immiscible

Unstable FlowMiscible vs. Immiscible

15

Immiscible Vertical Flow

16

Shock velocity

Fractional Flow

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

20

CO2 Gravity Current

Finite plume: How fast does it spread?

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

*

*

==

22

Regime Diagram

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

25

Sloping Aquifer

Slope: 2 degrees

26

Residual Trapping

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

41M = 4, 2, 0.5, 0.25, 0.1

SPM: 1D Buckley-Leverett

Quadratic Kr

42

2D, Two-Phase, Homogeneous

PVI=0.5 PVI=0.7

Before Breakthrough After Breakthrough

Particle Distribution

43

2D, Two-Phase, Heterogeneous

Quadratic Kr

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

48

Posters

1. CO2 Sequestration Capabilities in GPRS

2. Algebraic Multiscale Formulation for Compressible Multiphase Flow

3. Miscible Convection in Saline Aquifers

4. CO2 Gravity Currents and Residual Trapping in Saline Aquifers

5. Particle Tracking Method for Nonlinear Multiphase Flow