discretization and grid blocks ntnu author: professor jon kleppe assistant producers: farrokh shoaei...

DISCRETIZATION AND GRID BLOCKS

NTNU

Author: Professor Jon Kleppe

Assistant producers:

Farrokh Shoaei

Khayyam Farzullayev

Discretization and Grid Blocks

REFERENCES ABOUT EXIT

Contineous and discrete systems

Constant grid block sizes

Variable grid block sizes

Time discretization

Numerical formulations

Sensitivity to number of grid blocks

Sensitivity to Time step

Capillary and viscous forces

Upscaling

QUESTIONS

Discrete system: Continuous system:


?t

p

K

C

x

p

2

2






Time discretization





Upscaling

QUESTIONS

For i=2 to N-1:

Summation of forward and backward equations give us:


...,'''!3

,''!2

,'!1

,,32

txPx

txPx

txPx

txPtxxP

...,'''!3

,''!2

,'!1

,,32

txPx

txPx

txPx

txPtxxP

2

211

2

2 2xO

x

PPP

x

P ti

ti

ti

t

i

i-1 i i+1

x

x

i-1 i i+1i-1 i i+1

Backward expansions of pressure:

Forward expansion of pressure:






Time discretization





Upscaling

QUESTIONS

Solving backward equation and Eq. II give us:

For i=N:

For i=1:

...,'''!3

2/,''

!2

2/,'

!1

2/,,0

32

txPx

txPx

txPx

txPtxP

22

12

12

2

)(4

323

xOx

PPP

x

P Lttt

...,'''!3

2/,''

!2

2/,'

!1

2/,,

32

txPx

txPx

txPx

txPtNxP

Eq. I

Solving forward equation and Eq. I give us:

21

x

PL

N-1 N

x

PR

22

12

2

)(4

323

xOx

PPP

x

P RtN

tN

t

N

Eq. II






Time discretization





Upscaling

QUESTIONS


Finer description of geometry More realistic grid system

Better accuracy in areas of rapid changes in pressures and saturations

Specially useful in the neighborhood of production and injection wells

i-1 i i+1

x xi+1xi-1

The Taylor expansions:

...

!3

2/)(

!2

2/)(

!1

2/)( 31

211

1 iii

iii

iii

ii Pxx

Pxx

Pxx

PP

...

!3

2/)(

!2

2/)(

!1

2/)( 31

211

1 iii

iii

iii

ii Pxx

Pxx

Pxx

PP






Time discretization





Upscaling

QUESTIONS

The flow term :

22/12/1

)()(

)( xOx

xP

xfxP

xf

x

Pxf

x i

ii

i

Where :

)()( 12

1

1

2/1

xOxx

PP

x

P

ii

ii

i

)()( 12

1

1

2/1

xOxx

PP

x

P

ii

ii

i

For i=2 to N-1 :

)()()(

)(2)()(

)(2

)( 1

12/1

1

12/1

xOx

xxPP

xfxxPP

xf

x

Pxf

x i

ii

iii

ii

iii

i

Due to the different block sizes, the error terms are of first order only.

Flow equation for i=1 and i=N depends to boundary conditions.






Time discretization





Upscaling

QUESTIONS

x1 x2

1 2PL

Boundary conditions

Pressure condition at the sides of slab:

)(2/)( 1

1

2/1

xOx

PP

x

P L

)()(

)()(2

)(

)()(2

)(1

1

12/11

12

122/11

1

xOx

x

PPxf

xx

PPxf

x

Pxf

x

L

)()()(

)(2)()(

)(2

)( 1

12/12/1

xOx

xxPP

xfxPP

xf

x

Pxf

x N

NN

NNN

N

NRN

N

Same for pressure at the right hand side:

For i=1:

For i=N:






Time discretization





Upscaling

QUESTIONS

Flow rate specified at the sides of slab:

QL1 2

x1 x2

21

x

P

B

kAQL kA

BQ

x

PL

2/1

)()(

)()(2

)(1

12

122/11

1

xOx

kA

BQ

xx

PPxf

x

Pxf

x

L

)()(

)()(2

)( 1

12/1

xOx

xx

PPxf

kA

BQ

x

Pxf

x N

NN

NNNR

N

Same for flow rate at the right hand side:

For flow rate at the left hand side:






Time discretization





Upscaling

QUESTIONS

Time discretization

...),(!3

)(),(

!2

)(),(

!1),(),(

32

ttxP

tttxP

tttxP

tttxPtxP

Expansion forward:

...),(!3

)(),(

!2

)(),(

!1),(),(

32

txP

ttxP

ttxP

ttxPttxP

Expansion backward:

)( tOt

PP

t

P ti

tti

tt

i

)( tOt

PP

t

P ti

tti

t

i






Time discretization





Upscaling

QUESTIONS

All parameters except pi at (t+Δt) are in time t and are known , so simply by solving equation you can find pi at (t+Δt).

Use the forward approximation of the time derivative at time level t.

The left hand side is also at time level t.

Solve for pressures explicitly.


t

PP

k

c

x

PPP ti

tti

ti

ti

ti

2

11 2

x

i-1 i i+1t

This formulation has limited stability, and is therefore seldom used.

i-1 i i+1t + Δt

Explicit method:






Time discretization





Upscaling

QUESTIONS

Use the backward approximation of the time derivative at time level t+Δt.

The left hand side is also at time level t+Δt.

Solve for pressures implicitly.

t

PP

k

c

x

PPP ti

tti

tti

tti

tti

2

11 2

x

i-1 i i+1t

i-1 i i+1t + Δt

A set of N equations with N unknowns, which must be solved simultaneously. For instance using the Gaussian elimination method.

This formulation is unconditionally stable.

Implicit method:






Time discretization





Upscaling

QUESTIONS

Use the central approximation of the time derivative at time level t+Δt / 2.

The left hand side is also at time level t+Δt / 2.

t

PP

k

c

x

PPP

x

PPP ti

tti

tti

tti

tti

ti

ti

ti

211

211 22

2

1

The resulting set of linear equations may be solved simultaneously just as in the implicit case.

The formulation is unconditionally stable, but may exhibit oscillatory behavior, and is seldom used.

Crank-Nicholson method:






Time discretization





Upscaling

QUESTIONS

SWir

100 grid blocks:

40 grid blocks:

20 grid blocks:

10 grid blocks:

Sensitivity to number of grid blocks1 i 10

1 i 20

1 i 40

1 i 100

1020

40

100

SW

X / L

The more grid blocks we have, the smaller are the blocks sizes (Δx), smaller is the numerical dispersion because the discretization error is proportional to Δx2 .

1-Sor

0

1

0 1






Time discretization





Upscaling

QUESTIONS

0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1

0 10 20 30 40 50 60 70 80 90 100

Sensitivity to time step

Δt = 20 sec

Δt = 10 sec

Δt = 1 sec

The smaller are the time steps (Δt), the smaller is the numerical dispersion due to the discretization where the error is proportional to Δt.

SWir

SW

X / L

1-Sor






Time discretization





Upscaling

QUESTIONS

Capillary and viscous forces pressure difference across a grid block will be directly proportional to the size of this

( in the flow direction ).

The direct effect of capillary pressure will therefore often be dominating in a core-sized grid block while it is negligible in a full field simulation formation scale grid block.

Core plug:

Simulation grid block:

Capillary forces (capillary endpoint pressure)

Viscous forces (viscous pressure drop)10 cm

0.01

bar

production injection

0.7

bar

150 m

production injection

21 bar

0.7

bar






Time discretization





Upscaling

QUESTIONS

Geological models may contain millions of grid blocks representing geologically interpolated data (geostatistical realizations).

Numerical simulators cannot handle this level of detail due to cost limitations (applicable with less than millions of grid blocks).

The magnitude of the difference between fine and coarse scales is very significant.

The key problem is how to obtain effective input for the numerical flow simulator from data on finer scales.

This process is called upscaling.

Upscaling






Time discretization





Upscaling

QUESTIONS

Laminar sclae

grid blocks: 1.5 * 0.5 m

core plug

Formation scale

grid blocks: 60 * 5 m

Formation scalegrid blocks: 12 * 2.5 m

Formation scalegrid blocks: 1.5 * 0.5 m

High permiability Low permiability






Time discretization





Upscaling

QUESTIONS

Fine grids and coarse grids:

Fine grids:

Coarse grids:






Time discretization





Upscaling

QUESTIONS

Cij

Fij

The permeability tensor of a porous medium is specified on each fine-scale cell Fij, and must be upscaled or homogenized over each coarse-scale or computational cell Cij

Permeability tensor:






Time discretization





Upscaling

QUESTIONS

Recovery

Time

Fine grid model with original relative permeability

Coarse grid model with upscaled relative permeability

Every single upscaling step is quality controlled during the upscaling by comparing recovery from the fine model with the recovery from the upscaled coarse gridded model incorporating the pseudo curves.

Quality control:






Time discretization





Upscaling

QUESTIONS

Questions

1. Use Taylor series to derive the following approximations (include error terms):

4. Write the discretized equation on implicit and explicit forms.

t

P

2

2

x

P

t

P

2

2

x

P

a) Forward approximation of

d) Central approximation of

b) Backward approximation of

c) Central approximation of (constant x)

(variable x)

2. Modify the approximation for grid block 1, if the left side of the grid block is

maintained at a constant pressure, PL.

3. Modify the approximation for grid block 1, if grid block is subjected to a constant flow

rate, QL.






Time discretization





Upscaling

QUESTIONS

References

Kleppe J.: Reservoir Simulation, Lecture note 3

EPS






Time discretization





Upscaling

QUESTIONS

Title: DISCRETIZATION AND GRID BLOCKS (PDF)

Author: Name: Prof. Jon Kleppe

Address:NTNU

S.P. Andersensvei 15A

7491 Trondheim

Website

Email

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discretization and grid blocks ntnu author: professor jon kleppe assistant producers: farrokh shoaei...

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