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Numerical Reservoir Simulation

Topic overview

Introduction

Developers

References

Differential equations

Gridding

Difference approximation

Discretization error

Stability Analyses

Reservoir equations

Reservoir performance

Topic Overview

An introduction to standard numerical solution techniques for reservoir flow equations.

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Topic overview

Introduction

Developers

References


Gridding



Stability Analyses

Reservoir equations


Introduction


for mass flow


for mass flow

Numerical Modell

Numerical Modell

Stability analyses

Gridding

Difference Approximation Discretization Error

Reservoir Performance

For more information click on the subject you want to learn more about.

Reservoir equationsReservoir equations

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Topic overview

Introduction

Developers

References


Gridding



Stability Analyses

Reservoir equations


Discretization Techniques

General partial differential equations for reservoir fluid flow must be discretized before they can be treated computationally.

The most common techniques are:

- finite differences- finite elements

We will in in this module learn about the finite difference technique.

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Topic overview

Introduction

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Gridding



Stability Analyses

Reservoir equations


Finite difference approximations are used in most commercial reservoir simulation software to solve fluid flow equations numerically.

Main steps in a discretization procedure:

- replace differential operators by algebraic ciexpressions- compute approximate solution at given points and iiispecified times

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Finite Differences

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Topic overview

Introduction

Developers

References


Gridding



Stability Analyses

Reservoir equations


Differential Equations for Mass Flow

)()](][

[

:

w

wwww

ww

rw

BS

tQdp

Bkk

water

)()](][

[

:

w

wwww

ww

rw

BS

tQdp

Bkk

water

Mass conservation equations for Black Oil models:

)()](][

[

:

o

oooo

oo

ro

BS

tQdp

Bkk

oil

)()](][

[

:

o

oooo

oo

ro

BS

tQdp

Bkk

oil

)(

)](][

[)](][

[

:

o

so

g

g

goosoo

rogg

gg

rg

BRS

B

S

t

QdpRBkk

dpB

kk

gas

)(

)](][

[)](][

[

:

o

so

g

g

goosoo

rogg

gg

rg

BRS

B

S

t

QdpRBkk

dpB

kk

gas

Next

Where Ql are sink/source term

Discretization Techniques

Topic overview

Introduction

Developers

References


Gridding



Stability Analyses

Reservoir equations


Reservoir Equations

)(

:

,,,,,

w

wt

kjikjiwww B

S

t

VqT

water

)(

:

,,,,,

w

wt

kjikjiwww B

S

t

VqT

water

)(

:

,,,,,

o

ot

kjikjiooo B

St

VqT

oil

)(

:

,,,,,

o

ot

kjikjiooo B

St

VqT

oil

)(

:

,,,,,

o

os

g

gt

kjikjigoosgg B

SRB

S

t

VqTRT

gas

)(

:

,,,,,

o

os

g

gt

kjikjigoosgg B

SRB

S

t

VqTRT

gas

For more information click on the equation you want to learn more about.

Discrete equations for Black Oil models for block i,j,k:

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Topic overview

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Gridding



Stability Analyses

Reservoir equations


)(

:

,,,,,

w

wt

kjikjiwww B

S

t

VqT

water

)(

:

,,,,,

w

wt

kjikjiwww B

S

t

VqT

water

Water Equation

The water equation consists of three parts; a flow term, a well term and an accumulation term.

For more information click on the term of the water equation you want to learn more about.

Flow term + well term = accumulation term

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Topic overview

Introduction

Developers

References


Gridding



Stability Analyses

Reservoir equations


wzzwzzwyywyywxxwxxww TTTT

water

:

wzzwzzwyywyywxxwxxww TTTT

water

:

Flow Term for Water

The flow term for water consists of three terms, one for each coordinate direction.

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For more information click on the term of the equation you want to learn more about.html

Topic overview

Introduction

Developers

References


Gridding



Stability Analyses

Reservoir equations


Flow Term for Water in x- direction

The x-part consists of two terms; one to compute flow to neighbour block in the positive direction and one for flow in the negative direction.

)()(

)()(

:

1,,

,1,

21

21

iwiwkjiww

rwx

iwiwkjiww

rwxwxwxx

zyxB

kk

zyxB

kkT

water

)()(

)()(

:

1,,

,1,

21

21

iwiwkjiww

rwx

iwiwkjiww

rwxwxwxx

zyxB

kk

zyxB

kkT

water

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For information on block boundaries, click on the textbox. html

Topic overview

Introduction

Developers

References


Gridding



Stability Analyses

Reservoir equations


Flow Term for Water in y- direction

The y-part consists of two terms; one to compute flow to neighbour block in the positive direction and one for flow in the negative direction.

)()(

)()(

:

1,,

,1,

21

21

jwjwkijww

rwy

jwjwkijww

rwywywyy

zxyB

kk

zxyB

kkT

water

)()(

)()(

:

1,,

,1,

21

21

jwjwkijww

rwy

jwjwkijww

rwywywyy

zxyB

kk

zxyB

kkT

water

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Topic overview

Introduction

Developers

References


Gridding



Stability Analyses

Reservoir equations


Flow Term for Water in z- direction

The z-part consists of two terms; one to compute flow to neighbour block in the positive direction and one for flow in the negative direction.

)()(

)()(

:

1,,

,1,

21

21

zwzwjikww

rwz

zwzwjikww

rwzwzwzz

yxzB

kk

yxzB

kkT

water

)()(

)()(

:

1,,

,1,

21

21

zwzwjikww

rwz

zwzwjikww

rwzwzwzz

yxzB

kk

yxzB

kkT

water

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Topic overview

Introduction

Developers

References


Gridding



Stability Analyses

Reservoir equations


Well Term for Water

NextBackUp

Specification are different for production and injection wells.

Click here to see how the production term for water is given.

water

Topic overview

Introduction

Developers

References


Gridding



Stability Analyses

Reservoir equations

Reservoir performance )()(

:

, welliiww

rwiiw pp

B

kWIq

water

)()(

:

, welliiww

rwiiw pp

B

kWIq

water

Srr

khWI

w

ei

)ln(

2

Up

Well Equations for Black Oil Model

Pwell = pressure in the well

Topic overview

Introduction

Developers

References


Gridding



Stability Analyses

Reservoir equations


Srr

khWI

w

ei

)ln(

2

)()(

:

, welliioo

roiio pp

B

kWIq

oil

)()(

:

, welliioo

roiio pp

B

kWIq

oil

Up



Topic overview

Introduction

Developers

References


Gridding



Stability Analyses

Reservoir equations


Srr

khWI

w

ei

)ln(

2

)()()()(

:

, welliioo

sroiwellii

gg

rgiig pp

B

RkWIpp

B

kWIq

gass

)()()()(

:

, welliioo

sroiwellii

gg

rgiig pp

B

RkWIpp

B

kWIq

gass

Up



Topic overview

Introduction

Developers

References


Gridding



Stability Analyses

Reservoir equations


Accumulation Term for Water

])()[()(

:

1,,,, n

w

wn

w

wkji

w

wt

kji

B

S

B

S

t

V

B

S

t

V

water

])()[()(

:

1,,,, n

w

wn

w

wkji

w

wt

kji

B

S

B

S

t

V

B

S

t

V

water

BackUp

The change of mass of water in block i,j,k during time t between step n and n+1 is given by:

html

Topic overview

Introduction

Developers

References


Gridding



Stability Analyses

Reservoir equations


Evaluation on Block Boundaries

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Topic overview

Introduction

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References


Gridding



Stability Analyses

Reservoir equations


Oil Equation

)(

:

,,,,,

o

ot

kjikjiooo B

S

t

VqT

oil

)(

:

,,,,,

o

ot

kjikjiooo B

S

t

VqT

oil

Up NextBack

For more information click on the term of the oil equation you want to learn more about.

Flow term + well term = accumulation term

The oil equation consists of three parts; a flow term, a well term and an accumulation term.

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Topic overview

Introduction

Developers

References


Gridding



Stability Analyses

Reservoir equations


ozzozzoyyoyyoxxoxxoo TTTT

oil

:

ozzozzoyyoyyoxxoxxoo TTTT

oil

:

Flow Term for Oil

The flow term for oil consists of three terms, one for each coordinate direction.

NextUp


Topic overview

Introduction

Developers

References


Gridding



Stability Analyses

Reservoir equations


Flow Term for Oil in x- direction

)()(

)()(

:

1,,

,1,

21

21

ioiokjioo

rox

ioiokjioo

roxoxoxx

zyxB

kk

zyxB

kkT

oil

)()(

)()(

:

1,,

,1,

21

21

ioiokjioo

rox

ioiokjioo

roxoxoxx

zyxB

kk

zyxB

kkT

oil

NextUp



Topic overview

Introduction

Developers

References


Gridding



Stability Analyses

Reservoir equations


Flow Term for Oil in y- direction

)()(

)()(

:

1,,

,1,

21

21

jojokijoo

roy

jojokijoo

royoyoyy

zxyB

kk

zxyB

kkT

oil

)()(

)()(

:

1,,

,1,

21

21

jojokijoo

roy

jojokijoo

royoyoyy

zxyB

kk

zxyB

kkT

oil

NextBackUp



Topic overview

Introduction

Developers

References


Gridding



Stability Analyses

Reservoir equations


Flow Term for Oil in z- direction

)()(

)()(

:

1,,

,1,

21

21

zozojikoo

roz

zozojikoo

rozozozz

yxzB

kk

yxzB

kkT

oil

)()(

)()(

:

1,,

,1,

21

21

zozojikoo

roz

zozojikoo

rozozozz

yxzB

kk

yxzB

kkT

oil

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Topic overview

Introduction

Developers

References


Gridding



Stability Analyses

Reservoir equations


Well Term for Oil

NextBackUp


Click here to see how the production term for oil is given.oil

Topic overview

Introduction

Developers

References


Gridding



Stability Analyses

Reservoir equations


Accumulation Term for Oil

])()[()(

:

1,,,, n

o

on

o

okji

o

ot

kji

B

S

B

S

t

V

B

S

t

V

oil

])()[()(

:

1,,,, n

o

on

o

okji

o

ot

kji

B

S

B

S

t

V

B

S

t

V

oil

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Topic overview

Introduction

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References


Gridding



Stability Analyses

Reservoir equations


Gas Equation

)(

:

,,,,,

o

os

g

gt

kjikjigoosgg B

SRB

S

t

VqTRT

gas

)(

:

,,,,,

o

os

g

gt

kjikjigoosgg B

SRB

S

t

VqTRT

gas

Up Back

For more information click on the term of the equation you want to learn more about.

The gas equation consists of a flow term for gas and dissolved gas, a well term and an accumulation term for gas and dissolved gas.

Flow terms + well term = accumulation terms

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Topic overview

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References


Gridding



Stability Analyses

Reservoir equations


gzzgzzgyygyygxxgxxgg TTTT

gas

:

gzzgzzgyygyygxxgxxgg TTTT

gas

:

Flow Term for Gas

Up Next

The flow term for gas consists of three terms, one for each coordinate direction.


Topic overview

Introduction

Developers

References


Gridding



Stability Analyses

Reservoir equations


Flow Term for Gas in x- direction

)()(

)()(

:

1,,

,1,

21

21

igigkjigg

rgx

igigkjigg

rgxgxgxx

zyxB

kk

zyxB

kkT

gas

)()(

)()(

:

1,,

,1,

21

21

igigkjigg

rgx

igigkjigg

rgxgxgxx

zyxB

kk

zyxB

kkT

gas

NextUp



Topic overview

Introduction

Developers

References


Gridding



Stability Analyses

Reservoir equations


Flow Term for Gas in y- direction

)()(

)()(

:

1,,

,1,

21

21

jgjgkijgg

rgy

jgjgkijgg

rgygygyy

zxyB

kk

zxyB

kkT

gas

)()(

)()(

:

1,,

,1,

21

21

jgjgkijgg

rgy

jgjgkijgg

rgygygyy

zxyB

kk

zxyB

kkT

gas

NextBackUp


For information on block boundaries, click on the textbox. (not active yet) html

Topic overview

Introduction

Developers

References


Gridding



Stability Analyses

Reservoir equations


Flow Term for Gas in z- direction

)()(

)()(

:

1,,

,1,

21

21

zgzgjikgg

rgz

zgzgjikgg

rgzgzgzz

yxzB

kk

yxzB

kkT

gas

)()(

)()(

:

1,,

,1,

21

21

zgzgjikgg

rgz

zgzgjikgg

rgzgzgzz

yxzB

kk

yxzB

kkT

gas

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References


Gridding



Stability Analyses

Reservoir equations


Flow Term for Dissolved Gas

Up Back

For more information click on the term of the equation you want to learn more about.

ozzozsz

oyyoysyoxxoxsxoos

TR

TRTRTR

gasdissolved

:

ozzozsz

oyyoysyoxxoxsxoos

TR

TRTRTR

gasdissolved

:

The flow term for dissolved gas consists of three terms, one for each coordinate direction.

Next

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Topic overview

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Stability Analyses

Reservoir equations


Flow Term for Dissolved Gas in x- direction

)()(

)()(

:

1,,

,1,

21

21

ioiokji

oo

srox

ioiokji

oo

sroxoxoxsx

zyxB

Rkk

zyxB

RkkTR

gasdissolved

)()(

)()(

:

1,,

,1,

21

21

ioiokji

oo

srox

ioiokji

oo

sroxoxoxsx

zyxB

Rkk

zyxB

RkkTR

gasdissolved

NextUp



Topic overview

Introduction

Developers

References


Gridding



Stability Analyses

Reservoir equations


Flow Term for Dissolved Gas in y- direction

)()(

)()(

:

1,,

,1,

21

21

jojokij

oo

sroy

jojokij

oo

sroy

oyoysy

zxyB

Rkk

zxyB

RkkTR

gasdissolved

)()(

)()(

:

1,,

,1,

21

21

jojokij

oo

sroy

jojokij

oo

sroy

oyoysy

zxyB

Rkk

zxyB

RkkTR

gasdissolved

NextBackUp



Topic overview

Introduction

Developers

References


Gridding



Stability Analyses

Reservoir equations


Flow Term for Dissolved Gas in z- direction

)()(

)()(

:

1,,

,1,

21

21

zozojik

oo

sroz

zozojik

oo

srozozozsz

yxzB

Rkk

yxzB

RkkTR

gasdissolved

)()(

)()(

:

1,,

,1,

21

21

zozojik

oo

sroz

zozojik

oo

srozozozsz

yxzB

Rkk

yxzB

RkkTR

gasdissolved

BackUp



Topic overview

Introduction

Developers

References


Gridding



Stability Analyses

Reservoir equations


Well Term for Gas

NextBackUp


Click here to see how the production term for gas is given.

gas

Topic overview

Introduction

Developers

References


Gridding



Stability Analyses

Reservoir equations


Accumulation Term for Gas and Dissolved Gas

BackUp

])()[(

)(

:

1,,

,,

n

o

os

g

gn

o

os

g

gkji

o

os

g

gt

kji

BSR

B

S

BSR

B

S

t

V

BSR

B

S

t

V

gas

])()[(

)(

:

1,,

,,

n

o

os

g

gn

o

os

g

gkji

o

os

g

gt

kji

BSR

B

S

BSR

B

S

t

V

BSR

B

S

t

V

gas


html

Topic overview

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Stability Analyses

Reservoir equations


Definition of Symbols

l = o,w,g s = x,y,z p = i,j,k ql,i,j,k =

Ql,i,j,k = = Sl =

Bl = [k] = k = l =

Vi,j,k =t =t =

Rs =

Rs =

sTls =

sls =

WIp =

pi =

pwell =

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Reservoir equations


Difference Approximations

kk xxfk

xxfxxfxfxxf )(!

1)(

!2

1)()()( 2

kk xxfk

xxfxxfxfxxf )(!

1)(

!2

1)()()( 2

Taylor series can be used to derive a difference formula for single and double derivates.

With these expansion we can deduce:

- first order approximation of f ’

- second order approximation of f ’

- second order approximation of f ’’

Taylor series of f(x+x) and f(x-x) are given by:

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First Order Approximation of f’

Rx

xfxxfxf

)()(

)(

Rx

xxfxfxf

)()(

)(

This difference formula is used for discretizing time

derivative in the mass equations

From the expansion of f(x+Δx) we get an expression for f’(x):

Up

Click on the box to see how the approximation changes when the step size is halved.

xx

From the expansion of f(x-Δx) we get an expression for f’(x):

Next

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Reservoir equations

Reservoir performance t

uuu

nnn

t

11

Difference Formula

A first order approximation of ut at the point n+1 is given by:

The time axis is divided into points at distance Δt:

tt

n

1

t

n

Up

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Reservoir equations


First Order Approximation of f’

Rx

xfxxfxf

)()()(

Rx

xxfxfxf

)()(

)(

From the serie f(x+Δx):

From the serie f(x-Δx):

2x

2x

Back

The step size reduction produces more accurate approximations.

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Reservoir equations


Rx

xxfxxfxf

2

)()()(

Adding expansion of f(x+Δx) and f(x-Δx) results in the approximations:

X X

Up

Second Order Approximation of f’

NextBack

Click on the box to see how the approximation changes when the time step is halved. html

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Reservoir equations


2

x2

x

Rx

xxfxxfxf

2

)()()(

The sum of f’(x) of the series f(x+Δx) and f(x-Δx):

Up

Second Order Approximation of f’

Back

Step size reduction

will produce

more accurate

approximations.

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Reservoir equations

Reservoir performance 2

2

)()(2)()( x

x

xxfxfxxfxf

)()(2)(2 xxfxfxxff

The sum of the Taylor series f(x+Δx) and f(x-Δx) is used to deduced a second order approximation of f’’:

This approximation is frequently used and the numerator is written:

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Second Order Approximation of f’’

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Difference Approximation

xx

i

1

x

i

xx

i

1

2

11 2

x

uuuu iii

ixx

Uxx can be approximated at each point i by the formula:

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Reservoir equations


Discretization Error

The order of a difference approximation can by analysed using Taylor expansions.

The discretization error approaches zero faster for a high order approximation then for a low order approximation.

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Gridding

A faulted reservoir

Well locations An imposed grid

Initial fluid distribution

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Reservoir equations


A Faulted Reservoir

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Reservoir equations


Well Locations

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An Imposed Grid

Up

- The ability to identify saturations and pressures ii at specific locations (existing and planned well i iiiilocations).

- The ability to produce a solution with the i iiiirequired accuracy (numerical dispersion and iiiigrid orientation effects).

- The ability to represent geometry, geology and iiiphysical properties of the reservoir (external iiiboundaries, faults, permeability distribution iiiincluding vertical layering).- Keep the number of grid blocks small in order to iiimeet requirements of limited money and time iiiavailable for the study.

Main criteria for grid selection:

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Reservoir equations


Initial Fluid Distribution

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Stability Analyses

Stable Unstable

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Stability Analyses

Reservoir equations


Stable

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Animation of the stable solution

4,02x

t

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Stability Analyses

Reservoir equations


Unstable

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Animation of the unstable solution

6,02x

t

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Reservoir equations


Reservoir Performance

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Developers

Made by students

Siril Strømme and Rune Simonsen

Stavanger university college

Informasjon på min web-side

http://www.ux.his.no/~hans-k

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