Coupled Thermo-Hydro- Mechanical Analysis Daniel Swenson Shekhar Gosavi Ashish Bhat Kansas State...
Transcript of Coupled Thermo-Hydro- Mechanical Analysis Daniel Swenson Shekhar Gosavi Ashish Bhat Kansas State...
Coupled Thermo-Hydro-Coupled Thermo-Hydro-Mechanical AnalysisMechanical Analysis
Daniel SwensonDaniel SwensonShekhar GosaviShekhar Gosavi
Ashish BhatAshish Bhat
Kansas State UniversityKansas State UniversityMechanical and Nuclear Engineering Mechanical and Nuclear Engineering
DepartmentDepartmentManhattan, KS, 66506, USAManhattan, KS, 66506, USAe-mail: e-mail: [email protected]@ksu.edu
ObjectiveObjective
To provide coupled thermal-hydraulic-To provide coupled thermal-hydraulic-mechanical analysis tools that enable mechanical analysis tools that enable quantitative understanding and prediction quantitative understanding and prediction of thermal effects on flow in the reservoir.of thermal effects on flow in the reservoir.
ApproachApproach
Couple deformation/stress analysis Couple deformation/stress analysis with TOUGH2with TOUGH2
Couple wellbore model with Couple wellbore model with TOUGH2TOUGH2
Apply these tools to the analysis of Apply these tools to the analysis of Coso injectionCoso injection
StatusStatus
Implemented one way (forward) couplingImplemented one way (forward) coupling Implemented back coupling effect on Implemented back coupling effect on
hydraulic properties (porosity and hydraulic properties (porosity and permeability) without full Jacobian terms.permeability) without full Jacobian terms.
Now implementing full Jacobian solutionNow implementing full Jacobian solution Expect to have working version first Expect to have working version first
quarter of 2005quarter of 2005
System Equations for Stress System Equations for Stress CouplingCoupling
Conservation EquationsConservation Equations• MassMass• EnergyEnergy• MomentumMomentum
Constitutive EquationsConstitutive Equations• Darcy’s law (Advective Flux)Darcy’s law (Advective Flux)• Fick’s lawFick’s law (Diffusive Flux)(Diffusive Flux)• Fourier law (Thermal)Fourier law (Thermal)• Terzaghi’s Principle (Effective Stress) Terzaghi’s Principle (Effective Stress)
Fluid Mass BalanceFluid Mass Balance
0
QM
tJI
0
Sr SQt
Svq
0
ttt
T
t
pS
KS
Qt
S
VT
S
r
q
Change in Hydraulic PropertiesChange in Hydraulic Properties PorosityPorosity
'
0 exp Mrr a
k
k
pp cc0
0
0
Capillary PressureCapillary Pressure
Permeability Permeability
1exp
0
0
ckk
DiscretizationDiscretization
Fluid Flow [IFDM] Fluid Flow [IFDM] TOUGH2 MeshTOUGH2 Mesh
nmm
nmnm
n
mnmnm
n
n
nnTnn
nn
S
nn
n
QqAV
dt
uAd
V
dt
d
dt
Td
dt
pSd
Kdt
Md
1
Discretization (Contd.)Discretization (Contd.)
Momentum [FEM]Momentum [FEM] Cartesian DualCartesian Dual
VV
VT
V
uV
u
SVa
V
dVpdV
dVTK
dVp
dV
dSdVdV
mBσ'B
ΔmB
ΔmB
uΔBDB
tNbNuΔBDB
0
TT
T
T
T
TTT
3
1
Dual MeshDual Mesh
TOUGH2 MeshTOUGH2 Mesh Cartesian DualCartesian Dual
TOUGH2 Cell Center FEM Node
Solution TechniqueSolution Technique
Newton-Raphson (TOUGH2)Newton-Raphson (TOUGH2)
pik
ni
pipi
pi
k
n xRxxx
R,
1
,1,
1
Jacobian RepresentationJacobian Representation
S
F
S
F
SSSF
FSFF
RR
up
JJJJ
Terms Coupling ,
Coeff. Elastic
Coeff. TOUGH2
SFFS
SS
FF
JJ
J
J
Solid-Fluid CouplingSolid-Fluid Coupling
Jacobian Modifications (Contd.)Jacobian Modifications (Contd.)
mnm
n
V Au
FSJ
n
V
n
V
V
m
n
m
uK
uu
''
Volumetric Strain (IFDM) Volumetric Strain (IFDM)
m
mnnmV uuA
S
F
SSSF
FF
R
F
S
p
JJ
FSJ R
u
J
nm
Motivation for Coupling of Motivation for Coupling of Wellbore ModelWellbore Model
Settings at Coso (EGS) siteSettings at Coso (EGS) site• Low permeabilityLow permeability• Significant drawdownSignificant drawdown• Presence of two-phase flow and multiple Presence of two-phase flow and multiple
feedzonesfeedzones
Our goal is to provide enhanced capability in Our goal is to provide enhanced capability in
TOUGH2 to-TOUGH2 to-• Better model flow in geothermal systems Better model flow in geothermal systems
containing inclined wells with multiple feedzonescontaining inclined wells with multiple feedzones• account for varying flowing bottomhole pressureaccount for varying flowing bottomhole pressure
HOLA wellbore SimulatorHOLA wellbore Simulator Multi-feedzone wellbore simulator for pure waterMulti-feedzone wellbore simulator for pure water
GWELL and GWNACL-extensions of HOLAGWELL and GWNACL-extensions of HOLA
Can handle steady state, one-dimensional flow Can handle steady state, one-dimensional flow
(single and two-phase) in the well with varying (single and two-phase) in the well with varying
well-radiuswell-radius
2 approaches :2 approaches :• Option 1 (Wellhead-to-Bottomhole) Option 1 (Wellhead-to-Bottomhole) • Option 2 (Bottomhole-to-Wellhead)Option 2 (Bottomhole-to-Wellhead)
Simulates both production and injectionSimulates both production and injection
BackgroundBackground Murray and Gunn (1993) – coupling between Murray and Gunn (1993) – coupling between
TETRAD and WELLSIMTETRAD and WELLSIM
Hadgu et al., (1995) – TOUGH2 and WFSAHadgu et al., (1995) – TOUGH2 and WFSA
Coupled wellbore flow option in TOUGH2Coupled wellbore flow option in TOUGH2
• tables are generated for each well that are tables are generated for each well that are
used for interpolation.used for interpolation.
• limited to single feedzonelimited to single feedzone
Some features of the coupled code are,Some features of the coupled code are,
• No change in TOUGH2 input fileNo change in TOUGH2 input file
• ‘‘H----’ type of record in GENER block indicates H----’ type of record in GENER block indicates
coupled simulationcoupled simulation
• Input file format for the well is in similar spirit of Input file format for the well is in similar spirit of
HOLAHOLA
• Wellhead pressure as a time-dependent tabular Wellhead pressure as a time-dependent tabular
datadata
• Shut-in/Flowing optionShut-in/Flowing option
Coupling of HOLA with Coupling of HOLA with TOUGH2TOUGH2
Coupling of HOLA with Coupling of HOLA with TOUGH2 (Contd.)TOUGH2 (Contd.)
PROCEDURE:PROCEDURE:
i.i. Read input file Read input file
ii.ii. Obtain required reservoir parameters Obtain required reservoir parameters
iii.iii. Call HOLA at the start of each new time-stepCall HOLA at the start of each new time-step
iv.iv. A positive(/negative) flowrate at a feedzone in HOLA is A positive(/negative) flowrate at a feedzone in HOLA is
supplied as production(/injection) rate in the supplied as production(/injection) rate in the
corresponding source/sink element in TOUGH2corresponding source/sink element in TOUGH2
v.v. Enthalpy of a producing element is calculated in Enthalpy of a producing element is calculated in
TOUGH2, while for injection it comes from HOLATOUGH2, while for injection it comes from HOLA
vi.vi. Repeat steps (ii) to (v) for the next time-step with Repeat steps (ii) to (v) for the next time-step with
updated values of reservoir parameters.updated values of reservoir parameters.
Coupling of HOLA with Coupling of HOLA with TOUGH2 (Contd.)TOUGH2 (Contd.)
Minimal changes made to TOUGH2Minimal changes made to TOUGH2
Issues in HOLAIssues in HOLA
• Averaging of parameters in routine VINNA2Averaging of parameters in routine VINNA2• Relative permeability calculationsRelative permeability calculations• Instances of un-initialized variables being usedInstances of un-initialized variables being used• Division by zero Division by zero • Inclined wellsInclined wells• Hard-coded simulation parametersHard-coded simulation parameters
Sample ProblemSample Problem
Sample problem 5 from TOUGH2 user’s guideSample problem 5 from TOUGH2 user’s guide
• Well with inside diameter = 0.2 mWell with inside diameter = 0.2 m
• 500 m thick, two-phase reservoir500 m thick, two-phase reservoir
• Water at P = 60 bars, T=TWater at P = 60 bars, T=Tsatsat(P) = 275.5 ˚C, S(P) = 275.5 ˚C, Sgg = 0.1 = 0.1
• Wellhead pressure = 7 barsWellhead pressure = 7 bars
• feedzone depth =1000 mfeedzone depth =1000 m
• 1-D radial mesh, extends 10,000 m1-D radial mesh, extends 10,000 m
• Well Productivity Index = 4.64e-11 Well Productivity Index = 4.64e-11
• Simulation starts with a time-step of 1.e5 sec and ends Simulation starts with a time-step of 1.e5 sec and ends at time, 1.e9 sec (approx. 31.7 years)at time, 1.e9 sec (approx. 31.7 years)
Sample Problem (contd.)Sample Problem (contd.) Results obtained from the two runs plottedResults obtained from the two runs plotted These trends match with those obtained in TOUGH2 guideThese trends match with those obtained in TOUGH2 guide
20
30
40
50
60
70
80
1.00E+05 1.00E+06 1.00E+07 1.00E+08 1.00E+09
Time(sec)
Flo
w r
ate
(kg
/s)
or
Pre
ssu
re (
ba
rs)
1210
1220
1230
1240
1250
1260
1270
1280
1290
1300
En
thal
py
(KJ
/kg
)
Q(HOLA)Q(DELV)Pwb(HOLA)Pwb(DELV)h(HOLA)h(DELV)
Current/Future WorkCurrent/Future Work
Revisit the convergence methodology Revisit the convergence methodology
implemented in HOLAimplemented in HOLA
Extension to GWELL and GWNACLExtension to GWELL and GWNACL
Use the coupled code to better model the wells at Use the coupled code to better model the wells at
Coso (EGS) siteCoso (EGS) site
Finished first half of 2005Finished first half of 2005
AcknowledgementsAcknowledgements
Karsten Pruess and Jonny Rutqvist, LBNL.Karsten Pruess and Jonny Rutqvist, LBNL. Teklu Hadgu, Sandia National Teklu Hadgu, Sandia National
Laboratories.Laboratories. This work is supported by the U.S. This work is supported by the U.S.
Department of Energy, under DOE Department of Energy, under DOE Financial Assistance Award DE-FC07-Financial Assistance Award DE-FC07-01ID14186.01ID14186.
THANK YOUTHANK YOU
Mass Balance (Contd.)Mass Balance (Contd.) SolidSolid 01
1
SSS
tv
tt
T
t
p
KtV
TS
S
S
S
11
11
wherewhere
Solid Density Solid Density
tV
S
v
Mass Balance (Contd.)Mass Balance (Contd.) FluidFluid
0
ttt
T
t
pS
KS
Qt
S
VT
S
r
q
TOUGH2 Skeleton Solid Grains+ +
++
Energy BalanceEnergy Balance
GeneralGeneral v:σJv
HUt
U
Using Internal EnergyUsing Internal Energy• Neglecting conversion of KE to IENeglecting conversion of KE to IE
H
Cr
E
SS
E
ht
uuSJq
1
Momentum ConservationMomentum Conservation
GeneralGeneral Fσvvv
t
Static Equilibrium EquationStatic Equilibrium Equation• Neglecting inertial termsNeglecting inertial terms
Fσ
SS 1
Jacobian ModificationsJacobian Modifications
FFJ
CufSTfpfkfuSTpkI gg 4321 ,,,,,,,
termT2'
'
i
m
mi X
k
k
ffCfff
X
I
11
432
Fluid FlowFluid Flow
Individual TermIndividual Term
SSSSSF
FS
R
F
u
F
JJ
JFF RpJ
Stress EquilibriumStress Equilibrium
Jacobian Modifications (Contd.)Jacobian Modifications (Contd.)
SSJ
u
Vu
SV
tt
i
V
tti
a
itt
dVdSdV
dVu
uΔBDBtNbN
σBK
TTT
)1(
T)()1(
S
R
S
p
SSJ
JJ
RuJFF
SF
FSFF
Constitutive LawsConstitutive Laws Darcy’s Law (Advection)Darcy’s Law (Advection)
gkI
p
krr
Fick’s Law (Diffusion)Fick’s Law (Diffusion)
mvDJ
Fourier’s Law (Heat Conduction)Fourier’s Law (Heat Conduction)
Tm
H
C mJ
Fluid-Solid CouplingFluid-Solid Coupling
Jacobian Modifications (Contd.)Jacobian Modifications (Contd.)
SFJ
)1(
int
itt
ii XXF
R
Internal Forces – Dual MeshInternal Forces – Dual Mesh
lll
l AF int
S
R
u
F
JSF
JJ
R
p
JF
SSS
FSFF
Effective Stress LawEffective Stress Law
Stress-StrainStress-Strain
Effective Stress Effective Stress
pTKp T mσmmεDΔσ 0'31
TOUGH2 simulatorTOUGH2 simulator Numerical simulator for multi-phase fluid and Numerical simulator for multi-phase fluid and
heat flow in porous and fractured media.heat flow in porous and fractured media.
A well is represented in a simplified manner A well is represented in a simplified manner
Well on deliverability modelWell on deliverability model
• fixed bottomhole pressurefixed bottomhole pressure
• production rate is calculated as,production rate is calculated as,
)( wbr PPPIk
q
Coupled wellbore option Coupled wellbore option
Sample Problem (contd.)Sample Problem (contd.)
0
100
200
300
400
500
600
700
800
900
1000
0.0 1.0 2.0 3.0 4.0 5.0 6.0
Wellbore Pressure (MPa)
De
pth
(m
)
One Day
1.87 Year
32 Years