Multiphysical Numerical Simulation of CO2-EOR Processes Yu-Shu Wu Department of Petroleum Engineering Colorado School of Mines (CSM) Golden CO USA [email protected]
Multiphysical Numerical Simulation of CO2-EOR Processes
Yu-Shu WuDepartment of Petroleum Engineering
Colorado School of Mines (CSM)Golden CO [email protected]
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
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• 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
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
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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
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.
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