Development of a Reservoir Simulator with Unique Grid-Block System Master Division Student Paper Contest 2004 Harold Vance Department of Petroleum Engineering.

Development of

a Reservoir Simulator with

Unique Grid-Block System

Master Division

Student Paper Contest 2004

Harold Vance Department of Petroleum Engineering

Presentation Outline

Motivation

Problem Definition

Objectives

Approach

Results

Conclusions

Motivation

Not to scale

Diagonal Parallel

“Homogeneous Reservoir”

Homogeneous Reservoir

0

5000

10000

0 500 1000 1500 2000

Time (days)

Cu

m. O

il R

ec

ov

ery

SC

(b

bl) Diagonal Grid Parallel Grid

0

0.1

0.2

0.3

0.4

0.5

0.6

0 500 1000 1500 2000Time (days)

Wa

ter

Cu

t S

C

Diagonal Grid Parallel Grid

Motivation

Not to scale

Diagonal Parallel

K1 K2K1 >> K2

“Heterogeneous Reservoir”

Heterogeneous Reservoir

0

5000

10000

0 500 1000 1500 2000Time (days)

Cu

m. O

il R

ec

ov

ery

SC

(b

bl)


0

0.1

0.2

0.3

0.4

0.5

0.6

0 500 1000 1500 2000Time (days)

Wa

ter

Cu

t S

C


Motivation

Brand, Heinemann, and Aziz (1992) –

“In general, Grid Orientation Effect

cannot be overcome with

grid refinement.”

(SPE 21228)

Motivation

Todd et.al. (1972)

Yanosik & McCracken

(1979)

Pruess & Bodvarsson

(1983)

Shiralkar & Stephenson

(1987)

Shiralkar (1990)Brand et.al. (1991)

Sammon (1991)Chen & Durlofsky

(1991)

Ostebo & Kazemi (1992)

Mattax & Dalton (1990)Wolcott et.al.

(1996)

Fractures with Multiple Joint Sets

Courtesy of Imperial College

Fractures create high permeability anisotropy in rock masses!

Motivation

Motivation

General Darcy’s Law:

Permeability Tensor:

Simplified:

k

u~

00633.0Φ

μ

k0.00633u

~

zzzyzx

yzyyyx

xzxyxx

kkk

kkk

kkk

k~

zz

yy

xx

k00

0k0

00k

k~

Grid orientation and heterogeneity significantly affects the results of reservoir simulation

Problem Definition

Problem Definition

Problem Definition

We need a grid model that can We need a grid model that can incorporate incorporate permeability

anisotropy in multiple directions – in multiple directions –

a full tensor representation must a full tensor representation must be considered!be considered!

Objectives

Developing a 2-D, 3-Phase reservoir simulator using finite difference formulation

Reducing the grid orientation effects in a grid model

Creating a grid model that can be used to simulate multiple permeability directions

Approach

2-D, 3-Phase IMPES finite difference simulator using VBA

with unique grid model

2D,3-Phase

•Initial Condition

•Rock/Fluid Properties

Well ModelHGB Model

IMPES

Transmissibility Terms

Grid Numbering

Matrix Form

Matrix Solver

Pn+1, Son+1, Swn+1, Sgn+1

Program Validation

Well Constraints

3 4 5

1

2

3

4

I

J

1 21 2I

J

I

J

W E

N

S

NE

SESW

NW

Hybrid Grid Block (HGB) System

Hybrid Grid Block (HGB) System

Example Grid:

5 x 4

Total Number

of Grid Blocks = 61

I

J

1 2 3 4 5

1

2

3

4

IMPES MethodFinite Difference Equations

OilOil

WaterWater

GasGas

n

w

wp1n

w

wp1nnw B

SV

B

SV

t1

pa

n

g

gp

n

g

gpnng B

SV

B

SV

tpa

1

1 1

n

o

op

n

o

opnno B

SV

B

SV

tpa

1

1 1

x x x x x x xx x x x x x x x

x x x x x x xx x x x x x

x x x x x x x xx x x x x x x

x x x x xx x x x x

x x xx x x

x x xx x x

x x xx x x

x xx x

x xx x

Grid Numbering #1 & Matrix Form

1 of 3

1 2 3

4 5 6

7 8

15 9 10 16

13 14

17 11 12 18

Example: 3x2

2 4 6

9 11 13

10 12

1 3 5

8 14

15 16 17 18

x xx x x x x x x

x x xx x x x x x x x

x x xx x x x x x x

x xx x xx x x x x x xx x x x x

x x x x x x x xx x x x x

x x x x x x xx x x

x xx x x

x x xx x

2 of 3

7

Example: 3x2


x xx x x

x x x x x x xx x x

x x xx x x x x x x x

x x x x xx x x x x x x x

x x xx x

x x x x x x xx x x x xx x x x x x x x x

x x x x xx x x x x x x

x xx x xx x x x x x x x

x x x x xx x x x x x x x

x x xx x xx x x x x x x

x x xx x

3 of 3

1

2

3

4

5

6

1

8

10

7

9

11

12

18

19

14

13

15

16

17

20

21

22

24

25

23

1

2

3

4

5

6

8

10

7

9

11

12

18

19

14

13

15

16

17

20

21

22

24

25

23

Example: 3x3


Well Model

ooj r

m

kh

qBPP ln

2

Peaceman Well Model Well Model

(1983):(1983):

For square gridblock,

Δm

mro 208.0where,ro = effective wellbore radius

Well Model

Well Model for regular

polygon (after Palagi,1992):

j = neighbor of wellblock i

bij = side of polygon

dij = distance between gridpoints

Θij= angle open to flow

ijj

ij

ijj

o

db

θdb

expr

bij

dij

Ɵij

i

j

Results: Case#1

Model Dimension: Model Dimension: 640 ft x 640 ft x 10 ft640 ft x 640 ft x 10 ft

Permeability: 100mDPermeability: 100mDPorosity: 20%Porosity: 20%

Well Constraints:-Well Constraints:- Const. QConst. Qinjinj

Const. QConst. Qoo

Inj Prod

Maximum Material Balance Error = 4.2602E-05%

Results: Case#1

10 days 20 days

Contour Map

40 days

Results: Case #2

Same dataset, Same dataset, except:except:1 permeability permeability

directiondirectionKKmaxmax = 500 mD = 500 mD

KKminmin = 100 mD = 100 mD

Inj

Prod2

Prod1


Results: Case #2

10 days 20 days 40 days

Contour Map

Results: Case#2

Results: Case #3

Inj

Prod2

Prod1


Same dataset, Same dataset, except:except:3 permeability permeability

directionsdirectionsKKmaxmax = 500 mD = 500 mD

KKminmin = 100 mD = 100 mD

Results: Case #3

10 days 20 days 40 days

Contour Map

Results: Case #4


Homogeneous Homogeneous reservoirreservoir

1 injector1 injector4 producers4 producers

Results: Case #4

Pressure Distribution Chart

Conclusions

Grid orientation and and heterogeneity affects significantly the results of affects significantly the results of reservoir simulation (ie. water reservoir simulation (ie. water breakthrough times & recovery)breakthrough times & recovery)

A A full tensor representation must representation must be considered if reservoir flow be considered if reservoir flow performance is to be predicted performance is to be predicted accuratelyaccurately

Conclusions

Proposed HGB model is able to

reduce the grid orientation the grid orientation effectseffects

model model different sets of of permeability anisotropypermeability anisotropy

Future Application

Local Grid Refinement

Dr. David Schechter

Dr. Erwin Putra

U.S Department of Energy

Acknowledgement

THANK THANK YOUYOU

Development of

a Reservoir Simulator with

Unique Grid-Block System

Master Division

Student Paper Contest 2004

Harold Vance Department of Petroleum Engineering

Development of a Reservoir Simulator with Unique Grid-Block System Master Division Student Paper Contest 2004 Harold Vance Department of Petroleum Engineering.

Documents

motivation slide

j slide

unique grid model slide

days slide

3x2 slide

heterogeneous reservoir

matrix form slide

example grid