Shuyu SunEarth Science and Engineering program, Division of PSE, KAUST

Applied Mathematics & Computational Science program, MCSE, KAUST

Acknowledge: Mary F. Wheeler, The University of Texas at Austin

Abbas Firoozabadi, Yale & RERI; Joachim Moortgat, RERI

Mohamed ElAmin, Chuanxiu Xu and Jisheng Kou, KAUST

Presented at 2010 CSIM meeting, KAUST Bldg. 2 (West), Level 5, Rm 5220, 11:10-11:35, May 1, 2010.

Energy and Environment Problems

Single-Phase Flow in Porous Media

• Continuity equation – from mass conservation:

• Volumetric/phase behaviors – from thermodynamic modeling:

• Constitutive equation – Darcy’s law:

t

u qm

(T,P) (P)

u K

p

Incompressible Single Phase Flow

• Continuity equation

• Darcy’s law

• Boundary conditions:

],0(),( Ttxq u

],0(),( Ttxp K

u

p pB (x, t) D (0,T]

un uB (x, t) N (0,T]

DG scheme applied to flow equation

• Bilinear form

• Linear functional

• Scheme: seek such that

form

form

( , ) { }[ ] { }[ ]

[ ][ ]

h h h

D D h

e eE e eE T e E e E

ee ee e e

e e e E e

K K Ka p p p s p

K Kp s p p

h

n n

n n

ND e

e Be

e Be upK

sql

nform),()(

)( hkh TDp

)()(),( hkh TDvvlvpa

IIPG SIPG,0

NIPG0

DG-OBB0

IIPG0

NIPG DG,-OBB1

SIPG1

form

s

Transport in Porous Media

• Transport equation

• Boundary conditions

• Initial condition

• Dispersion/diffusion tensor

],0(),()()( * Ttxcrqccct

c

uDu

uc Dc n cBun t (0,T], x in (t)

Dc n 0 t (0,T], x out (t)

xxcxc )()0,( 0

)()()( uEIuEuIuD tlmD

DG scheme applied to transport equation

• Bilinear form

• Linear functional

• Scheme: seek s.t. I.C. and

houthh

hhh

Eee

e

e

Eee e

Eee e

Eee e

Eee e

TEE

ch

cqcc

csccccB

]][[][

]}[)({]}[)({)();,(

,

*

form

nunu

nuDnuDuuDu

))((),;(,

cMrcqccLinhEe

e eBw nuu

],0()(

))(;())(;,(),(

TtTD

MLMcBt

c

hr

hhhh

uu

)(),( hrh TDtc

Example: importance of local conservation

Example: Comparison of DG and FVM

Advection of an injected species from the left boundary under constant Darcy velocity. Plots show concentration profile at 0.5 PVI.

Upwind-FVM on 40 elements Linear DG on 20 elements

Example: Comparison of DG and FVM

Advection of an injected species from the left. Plots show concentration profiles at 3 years (0.6 PVI).

FVM Linear DG

Example: flow/transport in fractured media

Locally refined mesh:

FEM and FVM are better than FDfor adaptive meshes and complex geometry

Example: flow/transport in fractured media

Adaptive DG example

L2(

L2)

Err

or E

stim

ator

s

A posteriori error estimate in the energy norm for all primal DGs

2/1

2

)(

2/1

)(222

)(

hEELL

DG

LL

DG KcCcCΕ

uD

ELLB

ELLB

ELLB

ELLB

ELLBELLIEE

tRhRh

Rh

Rh

RhRh

2

))((0

2

))((0

2

))((0

2

))((1

2

))((1

2

))((

22

222

2222

2222

/2

1

2

1

1

2

1

2

1

Proof Sketch: Relation of DG and CG spaces through jump terms

S. Sun and M. F. Wheeler, Journal of Scientific Computing, 22(1), 501-530, 2005.

Adaptive DG example (cont.)

Ani

sotr

opic

mes

h ad

apta

tion

Adaptive DG example in 3DL

2(L

2) E

rror

Est

imat

ors

on 3

D

T=1.5

T=2.0

T=0.1 T=0.5 T=1.0

Two-Phase Flow Governing Equations

• Mass Conservation

• Darcy’s Law

• Capillary Pressure

• Saturation Summation Constraint

)( wcwnc SPPPP

wnpDgPk

ppp

rpp ,,

Ku

1 nw SS

wnpt

Spp

pp ,,0)(

u

DG-MFEM IMPES Algorithm – Pressure Equ

• If incompressible (otherwise treating it with a source term):

• Total Velocity:

• Pressure Equation:

• MFEM Scheme: – Apply MFEM – Two unknown variables: Velocity Ua and Water potential

wnpt

Sp

p ,,0

u

0 tu

cnwtcat KKuuu

nwt

p

rpp

k

,

ic

in

ia

iw

it KuK 11

wnpDgP ppp ,,

DG-MFEM IMPES Algorithm – Saturation Equ

• Solve for the wetting (water) phase equation:

• Relate water phase velocity with total velocity:

• Saturation Equation (if using Forward Euler):

• DG Scheme: – Apply DG (integrating by parts and using upwind on element

interfaces) to the convection term.

0

w

w

t

Su

at

waww f uuu

iwi

ia

iw

iwi

St

fSt

u1

• Relative permeabilities (assuming zero residual saturations):

• Capillary pressure

Reservoir Description (cont.)

2,,1, mSSSkSk wwem

wernmwerw

bars50 and5,,log)( cwwewecwec BSSSBSp

K=100md

K=1md

Comparison: if ignore capillary pressure …

Saturation at 10 years: Iter-DG-MFE

With nonzero capPres

With zero capPres

Saturation at 3 years

Iter-DG-MFE Simulation

Notice that Sw is continuous within each rock, but Sw is discontinuous across the two rocks

Saturation at 5 years

Iter-DG-MFE Simulation

Notice that Sw is continuous within each rock, but Sw is discontinuous across the two rocks

Saturation at 10 years

Iter-DG-MFE Simulation

Notice that Sw is continuous within each rock, but Sw is discontinuous across the two rocks

28

Compositional Three-Phase Flow

• Mass Conservation (without molecular diffusion)

• Darcy’s Law

gowPkr ,,,

gKu

gow

ii xc,,

,

uU

29

Example of CO2 injection

• Initial Conditions: C10+H2O(Sw=Swc=0.1), 100 bar,160 F.

• Inject water (0.1 PV/year) to 2 PV, then inject CO2 to 8 PV. Poutlet= 100 bar

• Relative permeabilities:

– Quadratic forms except nw=3.

– Residual/critical saturations:

• Sor = 0.40; Swc = 0.10; Sgc = 0.02

• Sgmax = 0.8; Somin = 0.2

• ; ; ;3.00

rwk 3.00rgk 3.00

rowk 3.00rogk

30

Example (cont.)

MFE-dG 0.1 PVI. MFE-dG 0.2 PVI. MFE-dG 0.5 PVI.

Example 3 (cont.)

nC10 at 10% PVI CO2 nC10 at 200% PVI CO2

32

Remarks for Multiphase Flow

• Framework has been established for advancing dG-MFE scheme for three-phase compositional modeling. In our formulation we adopt the total volume flux approach for the MFE.

• dG has small numerical diffusion

• CO2 injection – Swelling effect and vaporization– Reduction of viscosity in oil phase – Recovery by CO2 injection > Recovery by

C1 > Recovery by N2

EOS Modeling of Phase Behaviors

• PVT modeling: EOS– Peng-Robinson EOS – Cubic-plus-association EOS

• Thermodynamic theory• Stability calculation

– Tangent Phase Distance (TPD) analysis– Gibbs Free Energy Surface analysis

• Flash calculation– Bisection method (Rachford-Rice equation) – Successive Substitution – Newton’s method

,

I

I I

E

N V

,

II

II II

E

N V

,

I I

I I

E E

N V

,

II II

II II

E E

N V

,

I I

I I

E E

N V V

,

II II

II II

E E

N V V

1,

I I

I I

E E

N V

1,

II II

II II

E E

N V

Gibbs Ensemble Monte Carlo simulation

Particle displacements

Volume Change

Particle Transfer

Three Monte Carlo movements in simulation

The microstructure of the molecular models form the ab initio calculation

T-shaped pair of water molecules

The nearest neighbor interaction between the Water and Ethane

Microstructure from the ab initio calculation

Bond length(Å) Angle(degree) Hydrogen length(Å)OH(H2O) 0.9619OH(H2O) 2 0.9698 1.9321CH(C2H6) 1.0938CH(H2O----C2H6) 1.0940 (H2O) 105.06 (H2O) 2 105.28

107.5

HOH

HCHHOH

Water-ethane high pressure equilibria at T=523 K

EoS: Statistical-Associating-Fluid-Theory (SAFT)

Experimental data are from Chemie-Ing. Techn. (1967), 39, 816