Coupled Thermo-Hydro- Mechanical Analysis Daniel Swenson Shekhar Gosavi Ashish Bhat Kansas State University Mechanical and Nuclear Engineering Department.

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