A Novel Approach for Incorporation of Capillary and Gravity Into Streamline

Simulation Using Orthogonal Projection

Novermber, 2012Shusei Tanaka, Akhil Datta-Gupta and Michel J. King

Texas A&M University

Outline Background and Motivation

Orthogonal projection and operator splitting

Model Development Simulation workflow New formulation to incorporate capillary

and gravity effects Numerical Experiments

1D, 2D and SPE10 Conclusions

Split equation by physical mechanisms:

Operator Splitting

0

w

w utS

0

wt

w FutS

0~

wtw

w FuutS

DgpkFuFFFuu cowowtwwwtw Δ ~ ~

Anti-diffusive concave envelope

ww FF ~Anti-Diffusive flux

Includes anti-diffusive correction

DgpkFuFu cowowtww rr

Capillarity and Gravity

tuSplit

Convection

Difficult to implement anti-diffusive correction in multi-dimensional calculations

Construction depends on:• Fluid properties (p,T and composition)• Initial saturations• Relative permeability end points and tables

Capillarity not included in any commercial streamline simulator

Motivation

Anti-diffusive concave envelope

Shock construction of each grid-block is computationally expensive

Compute Pressure & Velocity Field Include Capillary Effects

Trace StreamlinesSolve 1D Convection EquationsInclude Capillarity and Gravity

Map Back Saturation to GridCalculate Diffusive Flux on Grid

Calculate Corrector Term

Predictor-Corrector Workflow

Pressure Equation• Governing Equation is given by 2-phase Black-Oil system

as

• Pressures to be solved as follows, second order term and capillary pressure derivative wrt. time is ignored.

woquSt

sc , , 0

01,

sc

wo

quSt

Time-of-Flight(TOF) : Travel time of a neutral tracer along streamlines

injectorproducer

, ,, ,

x y z

Inlet

dsx y zu

Streamline and Time-of-Flight

0

C

tC

Operator-Split:Convection on SL -> Diffusive flux on grid

0*

C

tC

)()( UHUfUt

tcowtwww

tcowtooo

cowwwotww

cowoowtoo

ww

oo

ugDpuFb

ugDpuFbH

gDpFbkuFb

gDpFbkuFbf

bS

bSU

Δ

Δ

Δ

Δ

21

21

2

2

, ,

Convection CompressionCapillary & Gravity

Water flow equation is calculated explicitly Most of the capillary and gravity effects are evaluated along streamline Remaining(orthogonal) diffusive flux is calculated on grids, after

solving 1D eq.

Final form of 1D saturation equation is be given as

1D Saturation Equations:A New Formulation

3D Saturation equation split into parallel and transverse flux terms

tu

Orthogonal Projection

twf u

wu

wu wtww uufu

0

w

w utS

0

wtw futS

0

w

w utS

• Specific example of operator splitting

Convection, Capillarity and Gravity along SL

Parallel component,calculate along Streamline

0

twwwwww ufbfbSbt

Saturation Equations Incorporating Capillarity and Gravity

Dgpuu

kFuuu

f cowt

t

t

oww

t

wtw

Δ22

Dgpuuu

uk

u cowt

tt

t

ow

tw

Δ 1 2

• 1D Flow equation including capillary and gravity

• Fractional flow with capillary and gravity (on SL)

• 3D corrector term (on grid) tu

twf u

wuwu

Water Velocity: Parallel and Orthogonal component

• Fractional flow with capillary and gravity with concave

• Component of water velocity orthogonal to total velocity:

DgpkFuFu cowowtww Δ

DgpkuuIFuuuIu cowttowwttw

Δ

ˆˆ ˆˆ

DgpkuuF

uuuf cow

t

t

t

oww

t

wtw

22

• Water phase velocity is given by

twf u

;

tuwu

wu

wtww uufu

cow

ttcowtp

ukupku ˆˆ

Permeability along Streamline• Permeability along streamline is evaluated as ‘penetrated

direction’, and are isotropic.

DukDku tzt

• Gravity term is always assumed by ‘kz’ regardless of streamline direction, to incorporate anisotropy.

Injection :: Water0.4 PVI – 4000 [Days], 1-step

xP

uuukf c

t

xx

t

owx

t

ww 2

Numerical Experiment: 1D

DgzP

uuu

kf c

t

zz

t

owz

t

ww

Δ2

Injection :: Water0.4 PVI – 4000 [Days], 1-step

2

o

w

Rock properties:

on

orwr

orwwcroro SS

SSSkk

11)(

wn

orwr

wrwoirrwrw SS

SSSkk

1)(

orwwnn

wnw

orwnwwrn

wwnhpccowt SSSSS

SSSSSSkcp

pc

pc

111

Case 1: Horizontal model Case 2: Vertical model

- Rel-Perm - Capillary

Water Saturation Distribution after 0.40PVI

Injection :: Water0.40 PVI – 4000 [Days]

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 0.2 0.4 0.6 0.8 1

Wat

er S

atur

atio

n

Normalized Distance

Commercial Simulator : 100 StepOrthogonal Projection : 1 StepOperator Split : 1 Step

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 0.2 0.4 0.6 0.8 1

Wat

er S

atur

atio

n

Normalized Distance

Commercial Simulator : 100 StepOrthogonal Projection : 1 StepOperator Split : 1 Step

Injection :: Water0.40 PVI – 4000 [Days]

Numerical Experiment: 1D

Horizontal model Vertical model

Orthogonal ProjectionFinite Difference

Operator Split(no correction)

Injection :: Water0.6 PVI – 60000 [Days]1,5,10,50,100 Time-steps

Production :: BHP (2900 psi)

Check the water saturation at production well block

Effect of Time-Stepping:2D Homogeneous Model

Pc & Convection

Pc (orthogonal)

Simulation time = 60000 days Split into 1,5,10, 50 or 100 time stepsPressure, then predictor, and then corrector solved at each time step

Water Saturation Distribution (1-step)(Along streamline, before correction term)

• Finite Difference

2D Homogeneous Model :Water Saturation Distribution

• Convection only • Convection + Capillary

Dispersion by capillary force is incorporated along streamline

• Operator Split (no correction)

Water Saturation at Production Block( 1,5,10,50,100 Time steps, 0.6 PVI, 60000 days)

• Orthogonal Projection

0.1

0.2

0.3

0.4

0.5

0.6

0 10000 20000 30000 40000 50000 600000.1

0.2

0.3

0.4

0.5

0.6

0 10000 20000 30000 40000 50000 60000

Finite Difference

1 Step5 Step

10 Step

100 Step

5-100 Step

1 Step

2D Homogeneous Model :Water Saturation at Well

Solution of OP converges with large time step size

Injection :: Water1.5 PVI – 1000 [Days]

Production :: BHP (2900 psi)

Check the water saturation at production well block

Effect of Time-Stepping:2D Cross-Sectional Model

Pc, Convection

Pc, Gravity

Water Saturation at Production Block( 1,5,10,50,100 Tsteps, 0.6 PVI, 1000 days)

• Orthogonal Projection

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 200 400 600 800 10000.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 200 400 600 800 1000

Finite Difference

1 Step

10 Step

50 Step100 Step

5 Step

1 Step

5 Step

10 Step

2D Cross-Sectional Model:Water Saturation at Well

• Operator Split (no correction)

Finite Difference

50-100 Step

Solution could not converge with 100 step size

Water Saturation Distribution (200 days, 5 step)Solution after corrector tem

• Orthogonal Projection• Finite Difference

Water Saturation Distribution

Injection :: Water0.5 PVI – 2000 [Days], 1-step

Permeability (500 md surface plot) Porosity (0.25 surface plot)

Application: SPE10 Model

• Orthogonal Projection• Operator Split (no correction)

• Finite Difference

SPE10: Water Saturation2000 Days, 1-step

Conclusion Have developed a new SL-based simulation

method to incorporate capillarity and gravity Computational advantages:

Can take large time steps without anti-diffusive corrections

Minimizes the saturation correction term Convergent solution demonstrated

Optimal time step strategies need to be developed