discret ization

7/27/2019 Discret Ization

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DISCRETIZATION OF

THE FLOW EQUATIONS

SIG4042 Reservoir Simulation


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FAQ References SummaryInfo

Constant GridBlock Size

Boundary Conditions

Spatial Discretization

Variable Grid

Block Size

Questions

Time Discretization

Explicit Formulation

Implicit Formulation

Crank-NicholsonFormulation

Nomenclature

Home

Handouts(pdf file)

Contents


Time Discretization

Crank-Nicholson Formulation



Questions

Constant grid block size

Variable grid block size

Boundary conditions
http://www.ipt.ntnu.no/~kleppe/SIG4042/note3.pdfhttp://www.ipt.ntnu.no/~kleppe/SIG4042/note3.pdfhttp://www.ipt.ntnu.no/~kleppe/SIG4042/note3.pdf


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Boundary Conditions


Variable Grid

Block Size

Questions

Time Discretization




Nomenclature

Spatial DiscretizationConstant grid block sizes

The approximation of the second derivative of pressure may be obtained by

forward and backward expansions of pressure:

...,'''!3

,''!2

,'!1

,,

32

txPx

txPx

txPx

txPtxxP

...,'''!3

,''!2

,'!1

,,

32

txPx

txPx

txPx

txPtxxP

2

2

11

2

2 2xO

x

PPP

x

P tit

i

t

i

t

i

it yields:

This approximation applies to the following grid system:

i-1 i i+1

x

Introduction

Finite difference approximations of the partial derivatives appearing in the

flow equations may be obtained from Taylor series expansions. We shallnow proceed to derive approximations for all terms needed in reservoirsimulation.

Continue

More


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Boundary Conditions


Variable Grid

Block Size

Questions

Time Discretization




Nomenclature

Spatial DiscretizationVariable grid block sizes

A more realistic grid system is one of variable block lengths, which will be the

case in most simulations. Such a grid would enable finer description ofgeometry, and better accuracy in areas of rapid changes in pressures andsaturations, such as in the neighborhood of production and injection wells.For the simple one-dimensional system, a variable grid system would be:

it yields:

i-1 i i+1

x xi+1xi-1

...

!3

2/)(

!2

2/)(

!1

2/)(3

1

2

111 i

iii

iii

iiii P

xxP

xxP

xxPP

Continue

...

!3

2/)(

!2

2/)(

!1

2/)(3

1

2

111 i

iii

iii

iiii P

xxP

xxP

xxPP

The Taylor expansions become (dropping the time index):

An important difference is now that the error term is of only first order,due to the different block sizes.

+


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Boundary Conditions


Variable Grid

Block Size

Questions

Time Discretization




Nomenclature

Spatial DiscretizationBoundary conditions

We have seen earlier that we have two types of boundary conditions,

Dirichlet, or pressure condition, and Neumann, or rate condition. If wefirst consider a pressure condition at the left side of our slab, as follows:

1 2

x1 x

2

PL

Morethen we will have to modify our approximation of the first derivative at the left

face, i=1/2, to become a forward difference instead of a central difference:

)(2/)( 1

1

2/1

xOx

PP

x

P L

The flow term approximation thus becomes:

)()(

)()(2

)(

)()(2

)(1

1

12/1

12

122/11

1

xOx

x

PPxf

xx

PPxf

x

Pxf

x

L

)()(

)()(2

)(

)()(2

)( 1

12/12/1

xOx

xx

PPxf

x

PPxf

x

Pxf

x N

NN

NNN

N

NRN

N

With a pressure PR specified atthe right hand face, we geta similar approximation forblockN:


7/20FAQ References SummaryInfo


Boundary Conditions


Variable Grid

Block Size

Questions

Time Discretization




Nomenclature

Spatial DiscretizationFor a flow rate specified at the left side (injection/production):

1 2

x1 x2

QL

More

we make use of Darcy's equation:

or:

Then, by substituting into the approximation, we get:

21

xP

BkAQL

kA

BQ

x

PL

2/1

)()(

)()(2

)(1

12

122/11

1

xOx

kA

BQ

xx

PPxf

x

Pxf

x

L




Boundary Conditions


Variable Grid

Block Size

Questions

Time Discretization




Nomenclature

Spatial DiscretizationWith a rate QRspecified at the right hand face, we get a similar

approximation for blockN:

More

For the case of a no-flow boundary between blocks iand i+1, the flow termsfor the two blocks become:

)()(

)()(2

)( 11

2/1

xOx

xxPPxf

kABQ

x

Pxf

x N

NN

NNNR

N

)()(

)()(2)(

1

12/1 xO

xxx

PPxf

x

Pxf

x iii

iii

i

)()(

)(

)(2)(121

12

2/111

xOxxx

PP

xfx

P

xfx iii

ii

ii




Boundary Conditions


Variable Grid

Block Size

Questions

Time Discretization




Nomenclature

Time discretizationTime discretization

We showed earlier that by expansion backward in time:

...),(!3

)(),(

!2

)(),(

!1),(),(

32

ttxPt

ttxPt

ttxPt

ttxPtxP

the following backward difference approximation with first order error term is obtained:

An expansion forward in time:

...),(

!3

)(),(

!2

)(),(

!1

),(),(32

txPt

txPt

txPt

txPttxP

yields a forward approximation, again with first order error term:

)( tOt

PP

t

P ti

tt

i

tt

i

)( tOt

PP

t

P titt

i

t

i

More




Boundary Conditions


Variable Grid

Block Size

Questions

Time Discretization




Nomenclature

Time discretizationFinally, expanding in both directions:

we get a central approximation, with a second order error term:

The time approximation used as great influence on the solutions of the equations.Using the simple case of the flow equation and constant grid size as example, wemay write the difference form of the equation for the three cases above.

...),(!3

)(),(

!2

)(),(

!1),(),(

2

3

2

2

2

2

2

2

2

tt

tt

tt

t txPtxPtxPtxPttxP

...),(!3

)(),(

!2

)(),(

!1),(),(

2

3

22

2

22

22

tt

tt

tt

t txPtxPtxPtxPtxP

22

tOt

PP

t

P titt

i

t

i

t

More




Boundary Conditions


Variable Grid

Block Size

Questions

Time Discretization




Nomenclature

Explicit FormulationExplicit formulation

Here, we use the forward approximation of the time derivative at

time level t. Hence, the left hand side is also at time level t, andwe can solve for pressures explicitly:

This formulation has limited stability,

and is therefore seldom used.

t

PP

k

c

x

PPP titt

i

t

i

t

i

t

i

2

11 2




Boundary Conditions


Variable Grid

Block Size

Questions

Time Discretization




Nomenclature

Implicit FormulationImplicit formulation

Here, we use the backward approximation of the time derivative at

time level tt, and thus left hand side is also at time level tt:

We have a set ofNequations with Nunknowns, which must be solvedsimultaneously, for instance using the Gaussian elimination method.

This formulation is unconditionally stable.

t

PP

k

c

x

PPP titt

i

tt

i

tt

i

tt

i

2

11 2




Boundary Conditions


Variable Grid

Block Size

Questions

Time Discretization




Nomenclature

Crank-Nicholson FormulationCrank-Nicholson formulation

Finally, by using the central approximation of the time derivative at time

level tt/2, and thus left hand side is also at time level tt/2:

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

The formulation is unconditionally stable, but may

exhibit oscillatory behavior, and is seldom used.

tPP

k

c

x

PPP

x

PPP titt

i

tt

i

tt

i

tt

i

t

i

t

i

t

i

2

11

2

11 22

2

1




Boundary Conditions


Variable Grid

Block Size

Questions

Time Discretization




Nomenclature

Questions1) Use Taylor series to derive the following approximations

(include error terms):

a) Forward approximation of

b) Backward approximation of

c) Central approximation of (constant x)

d) Central approximation of (variable x)Next

2) Modify the approximation for grid block1, if the left side of the grid

block is maintained at a constant pressure, PL.

3) Modify the approximation for grid block1, if grid block is subjected to

a constant flow rate, QL.

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

t

P

2

2

x

P

t

P

2

2

x

P


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Boundary Conditions


Variable Grid

Block Size

Questions

Time Discretization




Nomenclature

General information

About the author

Title: Discretization of the Flow Equations

Teacher(s): Jon Kleppe

Assistant(s): Szczepan Polak

Abstract: Derivation of finite difference approximations for all terms neededin reservoir simulation.

Keywords: discretization, spatial discretization, time discretization, explicitformulation, implicit formulation, Crank-Nicholson formulation

Topic discipline: Reservoir Engineering -> Reservoir Simulation

Level: 4Prerequisites: Good knowledge of reservoir engineering

Learning goals: Learn basic principles of Reservoir Simulation

Size in megabytes: 0.7

Software requirements: -

Estimated time to complete: 30 minutes

Copyright information: The author has copyright to the module
http://iptibm3.ipt.ntnu.no/~kleppe/http://polak.s.w.interia.pl/http://polak.s.w.interia.pl/http://iptibm3.ipt.ntnu.no/~kleppe/


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Boundary Conditions


Variable Grid

Block Size

Questions

Time Discretization




Nomenclature

FAQ


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Boundary Conditions


Variable Grid

Block Size

Questions

Time Discretization




Nomenclature

References

Aziz, K. and Settari, A.: Petroleum Reservoir Simulation, Applied SciencePublishers LTD, London (1979)

Mattax, C.C. and Kyte, R.L.: Reservoir Simulation, Monograph Series, SPE,Richardson, TX (1990)

Skjveland, S.M. and Kleppe J.: Recent Advances in Improved Oil RecoveryMethods for North Sea Sandstone Reservoirs, SPOR Monograph, Norvegian

Petroleum Directoriate, Stavanger 1992


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Boundary Conditions


Variable Grid

Block Size

Questions

Time Discretization




Nomenclature

Summary


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Boundary Conditions


Variable Grid

Block Size

Questions

Time Discretization




Nomenclature

About the Author

Name

Jon Kleppe

Position

Professor at Department ofPetroleum Engineering and

Applied Geophysics at NTNU

Address: NTNU S.P. Andersensvei 15A

7491 Trondheim

E-mail:

[email protected]

Phone:

+47 73 59 49 33

Web:

http://iptibm3.ipt.ntnu.no/~kleppe/
mailto:[email protected]?subject=e-learning%20modulehttp://iptibm3.ipt.ntnu.no/~kleppe/http://iptibm3.ipt.ntnu.no/~kleppe/mailto:[email protected]?subject=e-learning%20module

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