a novel approach for incorporation of capillary and gravity into streamline simulation using...
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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
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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
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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
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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
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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
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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
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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
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)()( 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
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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
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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
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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
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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.
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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
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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)
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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
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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
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• 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
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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
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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
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Water Saturation Distribution (200 days, 5 step)Solution after corrector tem
• Orthogonal Projection• Finite Difference
Water Saturation Distribution
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Injection :: Water0.5 PVI – 2000 [Days], 1-step
Permeability (500 md surface plot) Porosity (0.25 surface plot)
Application: SPE10 Model
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• Orthogonal Projection• Operator Split (no correction)
• Finite Difference
SPE10: Water Saturation2000 Days, 1-step
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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