Multiphysical Numerical Simulation of CO2-EOR Processes

Yu-Shu WuDepartment of Petroleum Engineering

Colorado School of Mines (CSM)Golden CO USAywu@mines.edu

Outline• Background of CO2 and CO2-EOR• Mathematical Model• Numerical Formulation• Results• Summary

Background - Total Primary Energy Demand (TPEM), World

https://www.iea.org/weo/

Background – Global CO2 Emissions4

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Background – U.S. CO2 Emissions

Background – Reason for Reducing CO2 Emissions?

STEO - Short-Term Energy Outlook

Background - Current Status of EOR

Mature technologies: Steam flooding CO2 flooding

Technologies with unrealized potential: Chemical EOR in-situ combustion Hydrocarbon miscible

Barriers: Lack in scientific breakthroughs or effective technologies Economics Politics

Application of CO2-EOR

8

• Typical incremental oil recovery by CO2 IOR/EOR is 5~25 % . • 93% of the CO2 projects in the world are in the U.S., contributing to 5% total US oil production.• CO2-EOR in the U.S. has steadily increased (134 total), but its growth has slowed down in the past

few years, primarily limited by accessible and affordable CO2 sources.

(Oil and Gas Journal EOR/Heavy Oil Survey 2014)

Background: IOR/EOR for Unconventional Reservoirs

• Low permeability; confined nano-pore space; large capillary pressure; highly heterogeneous/anisotropic

• Kerogen/organic matters; mechanically sensitive

Properties of unconventional reservoirs:

Issues in applying CO2 IOR/EOR to unconventional reservoirs:• Low Injectivity; fingering; poor swept volume/efficiency• Oil recovery is about 6% or lower• No effective or economically viable IOR/EOR approaches

developed or implemented in field-scale • Oil production relies mainly on primary recovery mechanisms • The only promising technology seems to be gas injection or huff-

n-puff, as demonstrated in lab and small-scale field tests

Thermal-Hydraulic-Mechanical (THM) Processes in CO2-EOR

Outline• Background of CO2 and CO2-EOR• Mathematical Model• Numerical Formulation• Results• Summary

Multiphysical Reservoir Simulation- Mathematical Model

Mechanical governing equation:

Fluid/Heat Governing Equation dd

= ∇⋅ +M F q

t

κκ

Energy Balance Equation

mean stress equation ( )3 2 , , ,kk T B ref L v kkP K T T G k x y zσ α β λ ε ε − + − = + =

Mass Balance Equation

𝑀𝑀𝑘𝑘 = 1 − 𝜙𝜙 𝐶𝐶𝑟𝑟𝜌𝜌𝑟𝑟𝑇𝑇 + 𝜙𝜙�𝑙𝑙𝑆𝑆𝑙𝑙𝜌𝜌𝑙𝑙𝑈𝑈𝑙𝑙

�⃗�𝐹𝛽𝛽 = −𝐾𝐾𝑎𝑎𝐾𝐾𝑟𝑟𝛽𝛽𝜌𝜌𝛽𝛽𝜇𝜇𝛽𝛽

𝛻𝛻𝑃𝑃 + 𝛻𝛻𝑃𝑃𝑐𝑐,𝛽𝛽 − 𝜌𝜌𝛽𝛽𝐠𝐠�⃗�𝐹𝑘𝑘 = �𝛽𝛽�⃗�𝐹𝛽𝛽𝑋𝑋𝛽𝛽

𝑘𝑘

𝑀𝑀𝑘𝑘 = 𝜙𝜙𝑆𝑆𝐿𝐿𝜌𝜌𝐿𝐿𝐿𝐿𝑘𝑘+𝜙𝜙𝑆𝑆𝐺𝐺𝜌𝜌𝐺𝐺𝐺𝐺𝑘𝑘 , . . . . . 𝑘𝑘 = 2, . . ,𝑛𝑛𝑛𝑛 + 1

𝐅𝐅𝑁𝑁+1 = −𝑘𝑘𝑡𝑡𝛻𝛻𝑇𝑇 + �𝑙𝑙ℎ𝑙𝑙𝐅𝐅𝑙𝑙

Mechanical constitutive relation： linear poro-thermo-elastic material

Navier’s equation： conservation of momentum

( ) ( ) 23 0T B L S S bP K T G u G u Fα β λ∇ + ∇ + + ∇ ∇ ⋅ + ∇ + =

Multiphysical Reservoir Simulation of CO2-EOR - Geomechanical Effects

Effective stress

Porosity and permeability

Mass accumulation

Capillary pressure

Outline

• Background of CO2 and CO2-EOR• Mathematical Model• Numerical Formulation• Results• Summary

Numerical Discretization and Formulation: MSFLOW_CO2

Integrated Finite Difference Method

10

l lk k k kn n nm nm n n

mn

tM M A F V qV

+ ∆ − − + = ∑

k k kM F qt

∂= ∇ ⋅ +

∂

n n n

kk k

V VM dV F nd q dV

t Γ

∂= ⋅ Γ +

∂ ∫ ∫ ∫

General formulation:

Newton’s method (gradient based searching)

Integral form:

Discretization:

Generalized Integral Finite-Difference Method:

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V

F

n

( )F F n F nΓ

∇ ⋅ = ⋅ Γ = ⋅∑∫ ∫ ijijVj

dV d A

Aij

Fij

Fij

Aij

Vj

Vi

xi

xj

di

dj

( )∂ ∂ ∂+ + = + + ∂ ∂ ∂

∑∫ yx zx x y y z z ijijV

j

FF F dV F n F n F n Ax y z

2 ijij ij ijV

j j i j

FFdV F d F A A

d dΓ

∆∇ = ∇ ⋅ Γ = ∇ ⋅ = ⋅

+∑ ∑∫ ∫ n n n n

( )2 2 2

2 2 22 2 2

∆ ∂ ∂ ∂+ + = + + ∂ ∂ ∂ +

∑∫ ijx y z ijijV

j i j

FF F F dV n n n Ax y z d d

Phase Equilibrium Calculation

Gibbs free energyG H TS= − Amount of non-PV energy

1

N

i ii

dG Vdp SdT dNµ=

= − +∑ Chemical potential / fugacity

Minimization

Equation of state

Equilibrium ratio,i krK

Flash calculation: P, T, Z x & α

with a reference phase

Equilibrium: Gibbs free energy is at the minimaFugacity of a component all phases are equal

, exp( ln )i i iiriri r

i ir i ir ir

x x ffK x f x fβ β β

ββ β

ϕϕ= = = −

Phase Equilibrium Calculation: Minimization of Gibbs Energy

Validation

Three phase three component system

C9+CO2+H2O

Comparison with experimental equilibrium ratio

C9+CH4+H2O

Calculation of Properties

Hydrocarbon phases..= , ,...

Z RTV G L

Pβ

β β = ....1 .= , ,G LVβ

βρ β =

1

.....= , , ,PN

i ii

S G L Aβ ββ

α ρβ

α ρ=

=

∑

Aqueous phase

Mole volume Mole density

Saturation Enthalpy ( )0 11

.....= , , = -275.....CN

i i ii

h x h G L h a a Tβ β β=

= +∑

Density Enthalpy Viscosity

Viscosity LBC (Lohrenz-Bray-Clark ) correlation

Mechanical properties

( ) ( )( )0 1 3p ref T refc P P T Tφ φ β= + − + −33

00

0

11

K K φ φφ φ

− = −

Porosity Permeability

1500 psi

15,000 psi120 F 240 F

Outline• Background of CO2 and CO2-EOR• Mathematical Model• Numerical Formulation• Results• Summary

Analysis of Thermal Effects

Permeability μD Porosity

Diameter μm

Permeability enhancement factor (1+b/P)

Permeability Pore radius Klinkerberg factor P & TPorosity

3000 psi

300 psi86 F 212 F

Analysis of Thermal Effects

2 cosCP

rγ θ

=

Capillary pressure:

( )( )

0 00

11c cP Pφ φφ φ

−=

− ( ) '

0= mar r e σφ φ φ φ −+ − ( )' B refK T Tσ β∆ = −

Capillary pressure Porosity Temperature Mechanical properties

Results: Cold CO2 Injection

Sg T

Young’s modulus 26 GPa

Poisson’s ratio 0.25 dimensionless

Rock permeability 1 md

Rock porosity 0.2 dimensionless

Biot’s coefficient 1.0 dimensionless

Grid block length 4 mInjection temperature 50 °C

Initial temperature 85 °C

Initial pressure 35.2 MPa

Production pressure 16.2 MPa

Initial mean stress 80.6 MPa

Residual gas saturation (Sgr) 0.1 dimensionless

Residual oil saturation (Sor) 0.1 dimensionless

Residual water saturation (Swr) 0.1 dimensionless

CH4+CO2 Puff-n-Huff

A typical well in Eagle Ford Shale Only a fraction of well is modelled Grids is refined near HF.

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Well geometry Fractional modelPerforated lateral length, ft 6840 X, ft 90Stage number 19 Y, ft 650Clusters per stage 4 Z, ft 100Cluster spacing, ft 90 HF half-length, ft 210Formation thickness, ft 100 HF height, ft 36

Numerical ExperimentsFive regions: propped HF, unpropped HF, NF in SRV, NF outside SRV, Matrix Different permeability, porosity, compressibility, Young’s modulus, Poisson’s ratio and Biot

coefficient for each region Different kr for fracture and matrix, kr in fractures has less residual saturation.

Six pseudo components are used. GOR 944 scf/stb (WinProp) v.s. 948 scf/stb (MSLOW_COM) above Pb

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Model Validation v.s. CMG-GEM without Pc and Geomechanics Effects

1. Depletion for 720 days, the accumulated CH4, CO2 reinjected. 2. Huff-n-puff with 91% CH4 &9% CO2: inj (150d)+soaking(30d)+prd(180d)

Oil rate Po & Sw at well grid

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Simulation Results

Case 1: Base case validated v.s. GEM; Case 2: Geomechanics effects (Geo);Case 3: nanopore confinement (Pc); Case 4: Pc coupled with geomechanicsAppreciable difference due to Geo but not Pc at early huff-n-puff.

Summary

• The world has seen significant enthusiasms and activities in CCUS and CO2-EOR in the last decade. It is a consensus that the best option for carbon management, currently or in the near future, is to combine the two technologies together, i.e., use CO2 for IOR/EOR for sequestration.

• Considerable effort and progress have been made at CSM to develop advanced CO2-EOR simulation tools and modeling studies, including geomechanics and thermal effects.

• A multiphase nonisothermal compositional model has been established for accurately modeling gas injection IOR/EOR processes in conventional and unconventional reservoirs.

• We are looking for collaboration for further CO2 and geo-energy related study, model improvement and application.

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Acknowledgements: Drs. Shihao Wang and Cong Wang

NREL

Thanks!

Questions?

Primary variables• Phase ratio

• Phase stability factor

• Concentration of each component in each phase

1

C

k ik ti

n nα=

=∑

ln irk

ik

ff

θ =

• Minimization of Gibbs Energy (π-1)

• The component concentration equation (π*C)

• The component concentration should be added up to 1 in each phase (π-1) ,

1 1 ,

( 1)0

1 ( 1)

k

j

Ci i kr

ki j j i jr

j r

z K eE

K e

θ

π θα= =≠

−= =

+ −∑

∑

0=+

=kk

kkkF

θαθα

, , ,1

[1 ( 1)] 0j ki k ik j i jr i i kr

jj r

D x K e z K eπ

θ θα=≠

= + − − =∑

,

1 ,1 ( 1)

k

j

i i krik

j j i jrj r

z K ex

K e

θ

π θα=≠

=+ −∑

Object functions

Phase Equilibrium Calculation

0k kα θ =

1,1,

ir

ik

equilibriumfnon eqiulibriumf

= < −

Equilibrium condition

• Fugacity

• Phase stability

Phase Equilibrium Calculation: Minimization of Gibbs Energy

Initialization

Estimate K by Wilson’s equation

, exp[5.373(1 )(1 )]ig ci cii go i

io

x P TKx P T

ω= = + −1=kα π

=0kθ

Newton’s method (gradient based optimization)

( )( )( )

12 1 2

ln 1 ln ln2 2 1 2

CN

j ijo oji i i

i

bPx a zf b bbP a RTz z bPx P b RT a bbRT zRT

=

+ − = − − − + − + +

∑

( )( )( )

12 1 2

ln 1 ln ln2 2 1 2

CN

j ijg oji i i

i

bPy a zf b bbP a RTz z bPy P b RT a bbRT zRT

=

+ − = − − − + − + +

∑

0 0

* * *

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lnT P

iAq i i iiAq

T P

g h vdT dP aRT RT RTRTµ

= − + +∫ ∫

SRK equation of state for oil phase

SRK equation of state for gas phase

Helgeson’s formation for water phase

Assume phase equilibrium and uniform distribution

Equation of State