2006-07 SPE Distinguished Lecture: Upscaling · Cell Permeability Upscaling: Laboratory and...

SPE DISTINGUISHED LECTURER SERIESis funded principally

through a grant of the

SPE FOUNDATIONThe Society gratefully acknowledges

those companies that support the programby allowing their professionals

to participate as Lecturers.

And special thanks to The American Institute of Mining, Metallurgical,and Petroleum Engineers (AIME) for their contribution to the program.

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Acknowledgements

• SPE International for the opportunity to participate in the 2006-07 Distinguished Lecturer Program

• BP America, Inc. for permission, and the Professional Recognition Program which has provided the time and resources to prepare and present this material

• Colleagues whose work is represented

• Local SPE chapters worldwide for their efforts in hosting these presentations

Upgridding and Upscaling:Current Trends and Future DirectionsDr. Michael J. King

Senior Advisor, Reservoir Modelling and SimulationBP America, Inc.

SPE 2006-07 Distinguished Lecturer

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Outline

• Introduction: Change of Scale & Upscaling

• Case Study: Magnus LKCF

– Validation and Analysis: What Went Wrong?

• Improved Upscaling: Understanding Permeability

– Boundary Conditions and Permeability Upscaling

– Transmissibility: Yes, Permeability: No

– Maintain the Well Injectivity & Productivity

• Magnus LKCF & Andrew Reservoir Case Studies

• Summary: What Works Well & What To Avoid?

• Future Trends:

– A Priori Error Analyses & Designer Grids

• Summary: Best Practice in Upscaling

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Introduction: What is Upscaling?

What is Upscaling?

• Assign “effective” properties to coarse scale cells from properties on fine scale grid

• Capture flow features of fine scale model

Why Upscale?

• Reduce CPU time for uncertainty analysis and risk assessment

• Make fine-scale simulation practical

─ geological models: ~10 -100 million cells

Resolution?

Image from Mike Christie

DW GOM

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Why Upscale?: CPU Time ReductionWaterflood Field Example

CP

U R

atio

(C

oar

se S

cale

/ F

ine

Sca

le)

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00

()

Uniform Layering CoarsenOptimum Layering CoarsenMCoarsen

Active Cell Ratio (Coarse Scale / Fine Scale)

Optimal Layer Coarsening

Uniform Layer Coarsening

Flexible 3D Coarsening

SPE 95759, King et.al.

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Upgridding and Upscaling: Context

Upgridding & Upscaling in the overall 3D Modelling Workflow

(After Roxar RMS)

• 3D Detailed Geologic Static Model

– Structure from well picks &/or seismic horizons

– Properties from well logs &/or seismicattributes &/or field performance data

– Geologic description from facies, analogues and field data

• Upscaled flow simulation model

– Performance prediction in the absence of dynamic data

– Starting point for a history match when dynamic data is available

• When done well, upscaling willpreserve the most important flow characteristics of a geologic model

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Why Upscale?: Length & Area

30 miles

30 miles

Lateral resolution of geologic and simulation grids are set by well

spacing

30 mile length of ACG reservoirs

with the London M25 loop used to

set the scale

Simulation Grid Cells: 200m x 200m or 100m x 100m

Geologic Grid Cells: 100m x 100m or 50m x 50m

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Kanaalkop: Tanqua Karoo basin, South AfricaDeepwater channel w/splay at top of photo

~15ft windmill

~10ft exposure

~250ft, which is about the size of a single cell in the areal direction of many simulation grids

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10ft thick exposure of channel…With 5 Components of a Bouma sequence

~10ft

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Why Upscale?: Thickness

Upscaling is dominated by loss of vertical resolution

Geologic grid will typically have 1 ft or 50 cm vertical resolution

Simulation grid may include only a single layer per geologic unit

600 ft section of a North Slope reservoir, with the 190 ft BP Anchorage office for scale

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Reservoir Zones, Well Logs & OutcropNo Vertical Exaggeration

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15 meters Geologist at Outcrop

30 geologic model layers

1-5 simulation model layers

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Outline










• Future Trends:



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Summary:What Works Well…

• Preserve connectivity and flow within the reservoir using flow based transmissibility upscaling

– Wide “Pizza Box” BC’s preserve flow tortuosity

– &/or Sealed Side BC’s preserve flow barriers

• Preserve flow between reservoir and wells using algebraic well index upscaling

– Preserves reservoir quality

• This combination of techniques has worked well within BP & similarly elsewhere in the industry

• Streamline calculations provide detailed validation based on pressures, sweep, and time of flight

– Validation after upscaling is always necessary

1

2

3

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Do We Have an Economic LKCF Waterflood Development?

LKCFLimit

OWC

MSMLimit

1 2 A- 9M5 :C4M1 6 :A4

1 km

M1 2 :A5-2700

LKCF

-2800

UKCF

-2900

-3000

MSM C-G

-3100

MSM A

-3200

B Shale

-3300

-3400Heather / Brent

-3500

-3600

-3700

1 2 A- 9M5 :C4M1 6 :A4

1 km

M1 2 :A5-2700

LKCF

-2800

UKCF

-2900

-3000

MSM C-G

-3100

MSM A

-3200

B Shale

-3300


-3500

-3600

-3700

1 2 A- 9M5 :C4M1 6 :A4

1 km

M1 2 :A5-2700

LKCF

-2800

UKCF

-2900

-3000

MSM C-G

-3100

MSM A

-3200

B Shale

-3300


-3500

-3600

-3700

Yellow = ChannelRed = MarginsBlue = Non-pay

64x64x450 = 1,843,200 cells 50mx50mx0.5m resolution

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Magnus LKCFWaterflood Development Study

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Cell Permeability Upscaling:Laboratory and Reservoir Model

• A laboratory coreflood

• In three dimensions, we have three numerical corefloods

– Coreflood follows the coarse cell shapes

– No flow side boundary conditions are the most common (others are possible)

QA

K PL

=µ∆

Darcy’s Law:

1 2

3 4

k*

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Streamlines in the Upscaled LKCF ModelHow Well Did 2x2x6 Upscaling Work?

• Time of Flight & Pressures after conventional 2x2x6 upscaling:

– Loss of 95% of effective permeability

– Loss of internal reservoir heterogeneity

Coarse Scale Time of Flight

Coarse Pressure

• 3D Streamlines, Time of Flight & Pressures calculated in the fine scale geologic model

– 2xInjectors & 2xProducers at a typical waterflood well spacing

– Fence diagram traced within the 3D geologic model

– Pressure constrained wells used to validate permeability

Fine Scale Time of Flight

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Cell Permeability UpscalingWhat Went Wrong?

600

200

0

0

500

600

0

300

300

0

0

300

0 0 600 0

KX Permeability0 100 200 300 400 500 600

0 0 0 0 600 600 600 600 600 0 0 0

0 0 0 0 600 600 600 600 600 0 0 0

0 0 0 0 600 600 600 600 600 0 0 0

0 0 0 0 600 600 600 600 600 0 0 0

0 0 0 0 0 600 600 600 600 600 0 0

0 0 0 0 0 0 600 600 600 600 600 0

300 300 300 0 0 0 600 600 600 600 600 0

300 300 300 0 0 0 0 600 600 600 600 600

300 300 300 300 300 0 0 600 600 600 600 600

300 300 300 300 300 300 0 0 600 600 600 600

300 300 300 300 300 300 0 0 600 600 600 600

300 300 300 300 300 300 0 0 600 600 600 600

• Sealed Side coreflood boundary conditions systematically expand barriers and reduce the continuity of pay

• Example 12x12=>4x4 (3x3 Upscaling):

• Continuous channel replaced by marginal sands

• Highly productive well replaced by poor producer

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Cell Permeability UpscalingStreamline Flow Visualization

• Each cell in isolation

• No cross-flow

• Equilibrium at cell faces

• Preserves & expands barriers

12x12 => 3x3

4x4 Upscaling Example

KX Cell Permeability KY Cell Permeability

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Cell Permeability UpscalingErrors & More Subtle Errors…

• Sealed Side Boundary Conditions do not adequately represent fluid flow in the fine scale model

– Reservoir quality is not preserved

– This is the most significant error

However, there are more subtle errors…

• Well Productivity (or Injectivity) is not preserved

– Well Index Upscaling

• Needless loss of spatial resolution

– Transmissibility Upscaling

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Outline










• Future Trends:



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Boundary Conditions andUpscaled Permeability - 1/2

• Upscale a simple sand / shale reservoir

• Sealed side BC’s expand barriers

• Open linear pressure BC’s allow barriers to leak

• “Pizza box”(Wide BC’s) allow global flow tortuosity

One Cell

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Boundary Conditions andUpscaled Permeability – 2/2

Question: Which permeability is right?

Answer:

• Wide “Pizza Box” (or tortuous) boundary conditions provide the best representation of fluid flow capacity, but…

• Sealed side boundary conditions preserve barriers.

– Barriers are often very important for modelling gas displacement, especially for vertical permeability

– They are also important in preserving channel margins

• Both answers are useful

– Use your judgement as engineers

• What is most important in your reservoir processes?

– Use both choices of boundary conditions as a sensitivity

• Mix and match horizontal and vertical treatments?

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Transmissibility Upscaling – 1/3Preserves Spatial Resolution

• Transmissibility can be calculated by direct upscaling instead of from the harmonic average of cell permeabilities

• “Link Permeability” is upscaled from cell center to cell center and has double the lateral resolution compared to cell permeability upscaling

• Harmonic average of a zero cell permeability is always zero

( ) ( )( ) ( ) 1

12121

2

+

+++ +

⋅⋅=

ii

iiii DXKXDXKX

DXKXDXKXATX

1

212121

2

+

+++ +

⋅=

ii

iii DXDX

KXATX

121 )()( ++ == iii MinusKXKXPlusKX

i 1+i

21+i

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Transmissibility Upscaling – 2/3KX Streamline Flow Comparisons

KXSealed

Cell

KXWide

Shifted

KXSealed Shifted

KXYWide No

Shift

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Permeability UpscalingDetermines Cell Properties

50 MD 50 MD 50 MD 50 MD50 MD

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Transmissibility Upscaling – 3/3Captures fine scale juxtaposition

0 MD 0 MD 38 MD 0 MD 50 MD50 MD

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Well Index UpscalingUsed to Preserve Reservoir Quality

• Well productivity / injectivity & sealed side coreflood permeability?

– Does not describe radial flow andlogarithmic pressure drop near a well

• Instead, use three (hypothetical)X, Y, and Z wells for each coarse cell

( )( )w

ZZ rr

HKYKXWI0ln

2 ⋅=

µπ

( )( )w

YY rr

HKZKXWI0ln

2 ⋅=

µπ

( )( )w

XX rr

HKZKYWI0ln

2 ⋅=

µπ

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Improved Upscaling:Well Index + Transmissibility

• Lack of pay continuity resolved through Well Index Upscaling

– Preserves injectivity and productivity of horizontal and vertical wells

– But, expands channels and removes barriers

• Contrast and barriers reintroduced through Transmissibility Upscaling

– Repeat in all three directions for 2x2x2=8-fold factor of improved flow resolution compared to cell permeabilities

KX Permeability0 100 200 300 400 500 600

0 0 0 0 600 600 600 600 600 0 0 0

0 0 0 0 600 600 600 600 600 0 0 0

0 0 0 0 600 600 600 600 600 0 0 0

0 0 0 0 600 600 600 600 600 0 0 0

0 0 0 0 0 600 600 600 600 600 0 0

0 0 0 0 0 0 600 600 600 600 600 0

300 300 300 0 0 0 600 600 600 600 600 0

300 300 300 0 0 0 0 600 600 600 600 600

300 300 300 300 300 0 0 600 600 600 600 600

300 300 300 300 300 300 0 0 600 600 600 600

300 300 300 300 300 300 0 0 600 600 600 600

300 300 300 300 300 300 0 0 600 600 600 600

600

467

200

200

533

600

0

300

300

200

67

300

0 400 600 0

600

467

200

200

533

600

0

300

300

200

67

300

0 400 600 0

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Coreflood Cell Permeability ORWell Index + Transmissibility Upscaling

• Coreflood Cell Permeability Upscaling

• Well Index + Transmissibility Upscaling KX Permeability0 100 200 300 400 500 600

600

200

0

0

500

600

0

300

300

0

0

300

0 0 600 00 0 0 0 600 600 600 600 600 0 0 0

0 0 0 0 600 600 600 600 600 0 0 0

0 0 0 0 600 600 600 600 600 0 0 0

0 0 0 0 600 600 600 600 600 0 0 0

0 0 0 0 0 600 600 600 600 600 0 0

0 0 0 0 0 0 600 600 600 600 600 0

300 300 300 0 0 0 600 600 600 600 600 0

300 300 300 0 0 0 0 600 600 600 600 600

300 300 300 300 300 0 0 600 600 600 600 600

300 300 300 300 300 300 0 0 600 600 600 600

300 300 300 300 300 300 0 0 600 600 600 600

300 300 300 300 300 300 0 0 600 600 600 600

600

467

200

200

533

600

0

300

300

200

67

300

0 400 600 00 0 0 0 600 600 600 600 600 0 0 0

0 0 0 0 600 600 600 600 600 0 0 0

0 0 0 0 600 600 600 600 600 0 0 0

0 0 0 0 600 600 600 600 600 0 0 0

0 0 0 0 0 600 600 600 600 600 0 0

0 0 0 0 0 0 600 600 600 600 600 0

300 300 300 0 0 0 600 600 600 600 600 0

300 300 300 0 0 0 0 600 600 600 600 600

300 300 300 300 300 0 0 600 600 600 600 600

300 300 300 300 300 300 0 0 600 600 600 600

300 300 300 300 300 300 0 0 600 600 600 600

300 300 300 300 300 300 0 0 600 600 600 600

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Outline










• Future Trends:



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LKCF Upscaling ValidationWell Index + Transmissibility

• 3D Streamlines & Time of Flight

• Comparison of:

– Fine Scale Model

– Coreflood Cell Perm Upscaling

– WI + Transmissibility Upscaling

Fine Scale Time of Flight


Coarse Pressure


Coarse Pressure

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Transmissibility Multipliers:Double the Spatial Resolution

• A transmissibility multiplier can represent a barrier without using a cell

• In contrast, zero vertical permeability prevents flow both up AND down and impacts flow in three layers

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Andrew Reservoir:Validation & Impact of Thin Barriers

Well Index + Transmissibility upscaling tracks

fine scale prediction &

early field performance

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Outline









• Summary: What Works Well & What to Avoid?

• Future Trends:



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Summary:What Works Well…

• Preserve connectivity and flow within the reservoir using flow based transmissibility upscaling

– Wide “Pizza Box” BC’s preserve flow tortuosity

– &/or Sealed Side BC’s preserve flow barriers

• Preserve flow between reservoir and wells using algebraic well index upscaling

– Preserves reservoir quality

• This combination of techniques has worked well within BP & similarly elsewhere in the industry

• Streamline calculations provide detailed validation based on pressures, sweep, and time of flight

– Validation after upscaling is always necessary

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Summary:What to Avoid…

• Flow based ‘coreflood’ upscaling for cell permeabilities

– Sealed side boundary conditions will not preserve flow tortuosity & will under-estimate reservoir quality

– Open linear pressure boundary conditions will not preserve reservoir barriers

• A single upscaling calculation cannot be used to preserve:– Reservoir quality

– Reservoir barriers

– Tortuosity of reservoir fluid flow around barriers

• Unfortunately, using coreflood permeability upscaling is the most common practice in the industry…

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Outline






– Transmissibility Yes, Permeability No




• Future Trends:



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Future Trends:A Priori Error Analyses & Designer Grids

• Wouldn’t it be nice to know if an upscaling calculation would be a good approximation before you performed the upscaling calculation?

• Sources of Upscaling Error

• Designer Grids: Upscale in the Simulator at Initialization

– Calculate Transmissibility and Pore Volume for Composite Cells

Assumption Source of Error (Missing Physics)

• Pressure equilibrium within the coarse cell

• Disconnected pay within the coarse cell will not be in equilibrium

• Fluid velocity is parallel to the pressure drop

• Flow may depend upon the transverse pressure drop on the coarse grid

• Single velocity within a coarse cell

• Distribution of multiphase frontal velocities replaced by a single value

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Error from Layer Coarsening:Flood Front Progression

• Error in the velocity distribution is introduced while upscaling

• Different fluid velocities are replaced by a single value

• F’(S) Kx/Phi is the frontal speed in each layer

– This is the property whose heterogeneity we will analyze

– Analysis applies to the net sands

• Vertical equilibrium within each coarse cell

FastSlow

MediumFastSlow

Medium

( ) ( )φXW KSF ⋅′ *

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Designer Grids within the Flow SimulatorStatic Boundary Conditions

• Source of A Priori Error: Multiphase frontal velocities are replaced by a single value

– Design simulation layering from 3D geologic model to minimize variation in local multiphase frontal velocities

249, 80% 336, 86%

0

10

20

30

40

50

60

70

80

90

100

0 200 400 600 800 1000 1200 1400 1600

Model Layers

%-H

eter

ogen

eity

0

2

4

6

8

10

12

14

16

18

20

RM

S R

egre

ssio

n - E

rror

Li & Beckner % Heterogeneity% Heterogeneity ; B-Variation% Heterogeneity: Uniform CoarsenDiagonal GuideSolution Total RMS RegressionSolution Weighted RMS RegressionTotal RMS RegressionWeighted RMS Regression

Number of Coarse Layers

%-H

eter

og

enei

ty

RM

S R

egre

ssio

n E

rro

rOptimal Layering

Uniform Coarsening: Not Efficient

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Designer Grids within the Flow SimulatorUpscale During Initialization (Static)

• General trend shows that uniform coarsening does not perform well

• “Optimal” (293 layers) is the best layering scheme

• Flexible 3D grid (MCOARSE) provides even better results

Tight Gas Layer CoarseningFine Scale Model 22x23x1715 (Geological Scenario 5)

0

5

10

15

20

25

30

0 200 400 600 800 1,000 1,200 1,400 1,600 1,800

Model Layers

Cum

. Gas

Pro

d. (B

CF)

Regular-CoarsenNextVar-OneStepNextVar-SequentialOptimal-12LOptimalLi-Map-12LLi-Ave-MaxLLi-Ave-12LMCOARSE

Fine Scale

Li and Beckner

UniformMCOARSE

Optimal

UniformCoarsening

OptimalLayering

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Layer Coarsening:Waterflood Example

Fine Scale124 Layers

Optimal22 Layers

7 LayersToo Coarse

22 Uniform LayersToo Coarse

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Waterflood Field Example:Oil Recovery and Watercut

• Optimal Simulation Model has 22 layers– 7 layers and 22 uniform layers are each too coarse

0%

5%

10%

15%

20%

25%

30%

0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00

FineScaleCoarsen_54Coarsen_22Coarsen_31Coarsen_19Coarsen_07

0%

5%

10%

15%

20%

25%

30%

0 2000 4000 6000 8000 10000 12000

FineScaleCoarsen_54Coarsen_22Coarsen_31Coarsen_19Coarsen_07

Oil Production

Time

Oil Production

PVINJ7 Layers7 Layers

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

0 2000 4000 6000 8000 10000 12000

FineScaleCoarsen_54Coarsen_22Coarsen_22UCoarsen_31Coarsen_19Coarsen_07

@

1760

1780

1800

1820

1840

1860

1880

1900

1920

1940

1960

1980

0 2000 4000 6000 8000 10000 12000

FineScaleCoarsen_22Coarsen_22U

Average Reservoir Pressure

Time

WaterCut

Time

7 Layers

22 Uniform Layers

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A Priori Error:Lack of Pressure Equilibrium

Assumption Source of Error (Missing Physics)

• Pressure equilibrium within the coarse cell

• Disconnected pay within the coarse cell will not be in equilibrium

• Fluid velocity is parallel to the pressure drop

• Flow may depend upon the transverse pressure drop on the coarse grid

• Single velocity within a coarse cell

• Distribution of multiphase frontal velocities replaced by a single value

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• Source of A Priori Error: Pressure equilibrium in the coarse cell is not present on the fine grid

– Design 3D simulation grid to prevent different sands from merging

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• General trend shows that uniform coarsening does not perform well

• “Optimal” (293 layers) is the best layering scheme

• Flexible 3D grid (MCOARSE) provides even better results

Tight Gas Layer CoarseningFine Scale Model 22x23x1715 (Geological Scenario 5)

0

5

10

15

20

25

30

0 200 400 600 800 1,000 1,200 1,400 1,600 1,800

Model Layers

Cum

. Gas

Pro

d. (B

CF)

Regular-CoarsenNextVar-OneStepNextVar-SequentialOptimal-12LOptimalLi-Map-12LLi-Ave-MaxLLi-Ave-12LMCOARSE

Fine Scale

Li and Beckner

UniformMCOARSE

Optimal

FlexibleCoarsening

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2-point Geostat Model, 100×100 → 10x10

Observations • Trans upscaling is better than k*• T* (open) > T* (restricted)• Linear pressure B.C. not good• Line/ point average good

λx = 1.0λy = 0.1σlogk = 1.735dx = 10.0 ftdy = 10.0 ft

-30% -20% -10% 0% 10% 20%ConstantPrePeriodicBCLinearPreConstantPre, vlConstantPre, lnConstantPre, ptPeriodicBC, vlPeriodicBC, lnPeriodicBC, ptLinearPre, vl

LinearPre, lnLinearPre, pt

Error to Fine-Scale Model Flow Rate, Qy = 7.02

restrictedopen

-15% -10% -5% 0%ConstantPrePeriodicBCLinearPreConstantPre, vlConstantPre, lnConstantPre, ptPeriodicBC, vlPeriodicBC, lnPeriodicBC, ptLinearPre, vl

LinearPre, lnLinearPre, pt

Error to Fine-Scale Model Flow Rate, QX = 47.8

restrictedopen

T*

K*

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Upscaling within the Flow SimulatorDynamic Boundary Conditions

• Source of A Priori Error: Fluid flow may depend upon the transverse pressure drop on the coarse grid

– Utilize actual well positions, flow rates and an iterative global solution on the coarse simulation grid to provide local pressure boundary conditions for the upscaling calculation, including the transverse pressure drop

Cell Permeability Transmissibility + Well PI Global Flow Rates

• 100x100x50 => 20x20x10 upscaling for a variogram-based fine scale model

• Material provided by Lou Durlofsky (Stanford) & Yuguang Chen (Chevron)

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Future Trends:Upscale in the Simulator (Static)

• 3x3x3 ‘coarsen’ used to reduce run-time

• Resolution re-introduced to preserveFault block boundariesResolution near wellsFluid contactsHeterogeneity via statistical measures

• More accurate flow simulation than with uniform coarsening

Workflow Implications

• Single ‘Shared Earth Model’used for both static anddynamic calculations

• Negligible time spent building coarse grid

• Extremely flexible grid design

• Simulation speed improvement comparable to model rebuild

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Outline






– Transmissibility Yes, Permeability No




• Future Trends:



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Summary: Best Practice in Upscaling

• Ensure that the fine and coarse grids are aligned– Many to 1 logical relationship is very important

• Check transport properties in initial geologic model– By Facies: NTG, Porosity, Horizontal Permeability, Kv/Kh ratio

• Conserve volumes when upscaling static properties & saturations– Bulk Rock Volume, Net Rock Volume, Pore Volume, Fluid Volumes

– Both 3DGeo => 3DSimulation and 1DLog => 1DGeo (blocked wells)

• When upscaling transmissibility (or permeability)– Preserve reservoir quality

– Preserve reservoir barriers

– Preserve flow around reservoir barriers

• Streamline-based flow validation after upscaling– Iteration: Is there a need to change resolution?

– Future trends: A Priori Error analysis & “Designer Grids”

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Summary: A Personal Literature Review

• Individuals whose work and questions have shaped my understanding of permeability & upscaling

• John Barker

• Karam Burns

• Dominic Camilleri

• Tianhong Chen

• Mike Christie

• Lou Durlofsky

• Don Peaceman

• Jens Rolfsnes

• Kefei Wang

• Chris White

• John K Williams

• Mike Zerzan

• Chris Farmer

• Kirk Hird

• Lars Holden

• Peter King

• Dave MacDonald

• Colin McGill

2006-07 SPE Distinguished Lecture: Upscaling · Cell Permeability Upscaling: Laboratory and...

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