1

SIMULATION OF DISCRETE FRACTURE NETWORKS USING FLEXIBLE VORONOI GRIDDING

Zuher SyihabDavid S. Schechter

2

Outline

• Problem Statement

• Objectives

• Introduction to Gridding Techniques

• Modeling Of Discrete Fracture Network (DFN) Using Voronoi Gridding

• Conclusion

3

PROBLEM STATEMENT

• Complex reservoir geometry (faults, fractures, etc)

• Limitation of the existing approach

• Fracture aperture measurements using X-Ray CT Scan.

• Capabilities of existing reservoir simulators

4

DUAL POROSITY MODEL

Idealization of fractured reservoirs (Warren and Root, 1963)

• Highly fractured media• Connected fractures• No flow occurs between matrix blocks

5

DUAL POROSITY MODEL - LIMITATIONS

• Not applicable for disconnected fractured media• Not suitable to model a small number of large-scale

fractures

Discrete Fracture Network (DFN)

Model

After SPE 79699 Karimi-Fahd, M., Durlofsky, L. J., and Aziz, K

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Fracture

Matrix

• Fractures are represented explicitly.

• Disconnected and isolated fractures

• Complex fractured porous media

• Difficult to be modeled with conventional rectangular grid system

• Current DFN model assumes the fracture apertures are uniform

DISCRETE FRACTURE NETWORK (DFN)

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OBJECTIVES• Develop a general flexible mesh generation

technique based on Voronoi diagram algorithm.• Developing a black-oil reservoir simulator to model

fractured and unfractured systems.• Honoring experimental work by incorporating

fracture aperture distribution into simulation model.• Performing simulation of a system with complex

intersecting fractures and fracture networks generated using fractal approach

Gridding Techniques

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• Globally Orthogonal Grid

• Corner Point Grid

• Locally Orthogonal

Grid (PEBI/VORONOI)

(a) point-dristributed (b) block-centered

History & Application of Voronoi Grid

• It was first applied by Heinrich into the reservoir simulation (1987).

• Heinmann named this grid as a PEBI grid (1989)

• Economides et. al applied PEBI grid to model horizontal wells (1991)

• Palagi studied the PEBI grid generation method (1994).

• Chong et. al had also shown that the kind of grid is able to reduce grid orientation effect (2004)

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10

VORONOI AND DELAUNAY TRIANGULATION

For a set S of points in the Euclidean plane, the unique triangulation DT(S) of S such that no point in S is inside the circumcircle of any triangle in DT(S).

The Voronoi grid is formed by the perpendicular-bisectors of the edges of the Delaunay triangles.

Delaunay Edges

Voronoi Edges

Circumcircle: a unique circle that passes through each of the triangles three vertices

MODELING DFN Workflow

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Fracture Geometry & Characterization

1. Geometry- Number of fracture- Size and orientation

2. Apetrure distribution

Populate points to generate fracture

(FRACTURE BLOCKS)

Populate points surrounding fractures

(MATRIX BLOCKS)

Add Points for fracture blocks in the

computational domain

Connection list and its properties

. . . . . . . .

. . . . . . . . . . . . . . . .

9w

matrix

matrix

w = fracture width

Flow Connection

AdditionalNodes forFracture

- Neural network- Outcrop- Kim’s network (fractal)

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Geometrical domain Computational domain

matrix

No Flow connection

w

matrix

matrix

Flow connection

MODELING DFN(Fracture Gridding)

(Line = fracture)

w = fracture width

Flow Connection

AdditionalNodes forFracture

Voronoi edge

12

2wk f

13

A B C A’ B’ C’

D E F D’ E’ F’

Geometrical domain Computational domain

APERTURE DISTRIBUTION AND VOLUME CORRECTION

d1 d1’ ‘

d2 d2’ ‘

df

d2’ ‘

The bulk volume of fracture segments can be computed based on given fracture apertures.

The bulk volume of the matrix block adjoining with the fracture should be corrected due to the volume taken by the fractures

Aperture distribution

Fracture Network & Voronoi AlgorithmMultiple-FractureSingle-Fracture

Voronoi Edges

Voronoi nodes

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1 2

3

4 5

6

7

8

F(1,2) Fracture-2

Fracture-1

F(2,1)

15

FRACTURES AND VORONOI DIAGRAM/PEBI (Example)

16

Kim’s Fracture Network(Fractal Geometry)

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Implemented using Visual basic (GUI for pre & post processor) and C++ (processor)

Fully implicit numerical method.

3D, 3-Phase black oil simulator.

Structured and unstructured grid systems.

Grid refinement features.- Radial-like grids- Hexagonal grids- Rectangular grids- Radom

Validated with analytical solution (Pressure Transient Analysis)

Compared with IMEX (CMG) for homogeneous & heterogeneous cases. (25 and up to 150,000 grid blocks in desktop PC).

DFN Simulator

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MATERIAL BALANCE EQUATIONS

Rate of accumulation =

n

e

p

n

e

p

B

SV

B

SV

t

)1(

1

Net flow rate =

NCon

ieiListConq

1)(..

,

e = Evaluated cell NCon = Number of connection of cell#e. Con. List(i) = the ith element in the connection list of cell#e

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nneiei eiei

MTq cei pZpei

Ncells

e

n

e

p

n

e

pNCon

i

neiListCon

neiListConeiListCon B

SV

B

SV

tMT

1

)1(

1

1)(.

1)(.)(.

1

11,

1,

1, n

ewn

egn

eo SSS

e

Con. List(i)

RESIDUAL FUNCTIONS

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Ncells

eosc

n

eo

op

n

eo

opN

i

n

eion

eioein

o qB

SV

B

SV

tMTr

1

)1(

1

111 1

Ncells

e

n

ew

wpnsw

n

eo

opnso

n

eg

gpN

i

n

eign

eigein

g B

SVR

B

SVR

B

SV

tMTr

1

1

1

1

1

1

1

111 ...1

scg

n

ew

wpnsw

n

eo

opnso

n

eG

Gp qB

SVR

B

SVR

B

SV

...

Ncells

ewsc

n

ew

wp

n

ew

wpN

i

n

eiwn

ewiein

w qB

SV

B

SV

tMTr

1

)1(

1

111 1

Wellbore ModelingCartesian Grid Block

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)( wfi PPJq

B

k r

w

or

rkh

Jln

2

)k/k( + )k/k(

k/k + k/k 0.28 = r 4/1

yxxy4/1

2yx

2xy

2/1yx

o

X

Peaceman’s Well model

Wellbore ModelingArbitrary Polygon

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Palagi’s Well model

ij

ij b

i

j

ij d

Center of grid

Ndb ij /tan/

NNdr ijo /tan

2exp

Regular Polygon

j ij

ijij

ijj

o

d

b

dd

b

r

)ln(

exp

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THE SIMULATOR(Implementation Technique)

jiwjigjio aaa ,,, ,,

Create Control Volume Objects

Connection List

Calculate flow coef.

PVT ID

Rock ID

etc

Connection typeVectorization

Solve the matrix usingSparse Matrix Solver (SparseLib++)

(BICG-STAB/GMRES/RI/BICG/CG)

Residual Error Checking

SIMULATION WORKFLOW

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GRID MODEL

Chapter 2 PVT Table(s)- Oil, Gas, Water

Rock Properties - Relative Permeability Curves - Capillary Pressure

Wells - Locations - Well type (producer/Injector)

PROPERTIES ASSIGNMENT - Constrains (Rate, BHP, etc)k L , k v and

INITIALIZATIONp i , s wi , s gi , s oi

DATA COLLECTION

t = 0

START

A

k = 0, n = 0

CONSTRUCT RESIDUAL FUNCTIONSFOR OIL, GAS & WATER

Eq.3.32. Eq.3.33 & Eq.3.34

PERTURB THE RESIDUAL FUNCTIONAND COMPUTE THE JACOBIAN, J

Eq.3.44

SOLVE THE LINEAR EQUATIONUSING SparseLib++ Solver

Eq.3.45

TEST THE SOLUTIONS

p k+1 , s wk+1 , s g

k+1 & s ok+1

Eq.3.41, Eq.3.42 & Eq.3.43

IS ABS(rok+1 ) <=TOL ?

IS ABS(rwk+1 ) <=TOL ?

YES IS ABS(rgk+1 ) <=TOL ? NOnext time step update fluid & rock properties

iter, k = 0 increase of iterations

time = end of simulation (tMAX)

A

n = n + 1t = t + tk = 0

k = k + 1

Update PVT and Rock properties

(k=k+1)

Update PVTand Rock

properties(n = n+1)

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VALIDATION AND COMPARISON STUDY

DFN simulation and analytical model (Pressure Transient Solution)

DFN simulation and IMEX on modified SPE-1 comparative study

DFN simulation and IMEX on heterogeneous case

DFN simulation and IMEX on unstructured grid case

DFN simulation and dual-porosity shape factor

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DFN SIMULATOR & ANALYTICAL MODEL(Constant Pressure Boundary)

p = 4790 psia

p =

479

0 ps

ia

p =

479

0 ps

ia

p = 4790 psia

k = 215.0 mdh = 100.0 ftpi = 4790 psia

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DRAWDOWN & BUILDUP DERIVATIVE PLOTS(Constant Pressure Boundary)

Pressure and Pressure Derivative Plot

0.1

1

10

100

0.1 1 10

t, hour

p-p(

0) a

nd p

' (ps

ia)

Pressure and Pressure Derivative Plot

0.1

1

10

100

0.1 1 10 100 1000

t, hour

p-p(

0) a

nd p

' (ps

ia)

Radial Flow Regime

Boundary Effect

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DFN & IMEX ON MODIFIED SPE-1 COMPARATIVE STUDY

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DFN & IMEX ON HETEROGENEOUS CASE

Comparison on Block Pressure Profile

CMG and DFN with 2,500 Gridblocks (25 X 25 X 4)

(Heterogeneous Case)

4550

4600

4650

4700

4750

4800

4850

0 20 40 60 80 100 120 140 160 180 200

Time, days

Pre

ss

ure

, ps

ia

DFN(1,1,1)

CMG (1,1,1)

DFN(25,25,1)

CMG(25,25,4)

Relative Error

Maximum : 0.087%

Average : 0.025%

KX / KY

(1,1,1)

(25,25,4)

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Block Pressure Profile

4680

4700

4720

4740

4760

4780

4800

0 20 40 60 80 100 120Time, days

Pre

ss

ure

, ps

ia

CMG

RZ-REGULAR

REGULAR

HEXA

TRIANGLE

IRREGULAR

DFN & IMEX ON UNSTRUCTURED GRID CASE

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DFNSIM AND DUAL-POROSITY SHAPE FACTOR

“shape factor” is the value to quantify the matrix-fracture drainage in the dual-porosity model.

Matrix-fracture drainage in the dual-porosity: *fm

m

b

ppk

V

qq

Fracture width, w 0.001 ft

Fracture spacing, Lx=Ly 100 ft

Fracture permeability, kf 10,000 md

Matrix Permeability, km 0.0001 md

Formation Volume Factor, Bo 1.0 RB/STB

Viscosity, o 0.7 cp

Bulk Volume, Vb 100x100x20 = 200,000 cu-ft

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DFNSIM AND DUAL-POROSITY SHAPE FACTOR

2.28

2L

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DFNSIM AND FRACTURE APERTURE DISTRIBUTION

Descriptions CASE 5.A1 CASE 5.A2

Grid dimension 33x33x1 33x33x1

Fracture spacing 1,220 ft 1,220 ft

Model width/ Length 5,380.4 ft 5,380.4 ft

Model thickness 100 ft 100 ft

Matrix permeability 50 md 50 md

Fracture

permeability

Constant

9,055 md

Log-normally distributed

24 md – 300 D (mean = 9,055 md)

Matrix porosity 0.25 0.25

Fracture porosity 0.5 0.5

Fluid properties SPE-1 SPE-1

Initial conditions SPE-1 SPE-1

Other rock

propertiesSPE-1 SPE-1

Producing rate Oil, 15,000 STB/D Oil, 15,000 STB/D

Minumum produce

BHP1,000 psia 1,000 psia

Injection rate Gas, 50 MMSCF/D Gas, 50 MMSCF/D

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0.0

2,000.0

4,000.0

6,000.0

8,000.0

10,000.0

12,000.0

14,000.0

16,000.0

0 1 2 3 4 5 6 7 8 9 10

BHP,

psi

a

Time, Years

Uniform fracture apertures

Log-Normally Distributed Fracture apertures

0.0%

10.0%

20.0%

30.0%

40.0%

50.0%

0 1 2 3 4 5 6 7 8 9 10

Reco

very

Fac

tor,

%

Time, Years

Uniform fracture apertures

Log-Normally Distributed Fracture apertures

DFNSIM AND FRACTURE APERTURE DISTRIBUTION

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DFNSIM AND ISOLATED FRACTURE NETWORK

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0

50

100

150

200

0.00 0.50 1.00 1.50 2.00

GO

R, M

SCF

/ST

B

Time, year

NO FRACTURE

ISOLATED FRACTURES

0.40

0.45

0.50

0.55

0.60

0.65

0.70

0.75

0.80

0.85

0.90

0.00 0.50 1.00 1.50 2.00

Oil

satu

rati

on, f

ract

ion

Time, year

NO FRACTURE

ISOLATED FRACTURES

DFNSIM AND ISOLATED FRACTURE NETWORK

SIMULATION ON FRACTAL DISCRETE FRACTURE NETWORK

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2

4

6

30

210

60

240

90

270

120

300

150

330

180 0

Rose Diagram of FDFN

-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

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SIMULATION ON FRACTAL DISCRETE FRACTURE NETWORK

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SIMULATION ON FRACTAL DISCRETE FRACTURE NETWORK

Numerical ParametersNo fracture, isolated and Connected Fractures

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Numerical Controls No Fracture Isolated Fractures Complex Fractures

Maximum residual error 1.0E-4 1.0E-4 1.0E-4

Max. Newton iteration 25 25 25

Max. linear solver iteration 40 40 140

Linear solver tolerance 1.0E-5 1.0E-5 1E-5

Time step 152 324 2,045

Newton iteration 976 6,576 34,285

Solver iteration 28,315 216,445 1,420,171

Solver failure 0 5 103

Time step cut 7 183 228

Simulation time 458 sec. 8,009 sec. 56,125 sec.

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CONCLUSION

• Dual Porosity models are not applicable for small scale and disconnected fractured media.

• The DFN simulator provides results in good agreement with commercial finite-difference simulators in the cases in which direct comparisons are possible.

• Fracture aperture distribution can be descritized using DFN model.

• DFN model using Voronoi grid system can be used for fractured and unfractured system.

• The aperture distribution plays very important role reservoir performance.

• Numerically, simulation on fractured systems, whether disconnected or connected, are very challenging. It requires an extensive amount of time to build the grid model and run the simulation.

• DFN simulator capability for multiple reservoir has been tested and it can be a potential tool for sensitivity studies.

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CONCLUSION