SUPRI-HW

Simulation of Naturally FracturedReservoirs with Dual Porosity Models

Pallav SarmaProf. Khalid AzizStanford University

2SUPRI-HW Nov. 20-21, 2003

Motivations

! NFRs represent at least 20%ofworld reserves, but difficult toproduce

! Unfeasible to model typicalmassively fractured NFRsthrough discrete fracturemodels

! Many limitations of existingdual porosity models

Circle Ridge Fractured Reservoir, Wyoming

3SUPRI-HW Nov. 20-21, 2003

The Dual Porosity Model

Matrix Continuum

Transfer Function

Fracture Continuum

4SUPRI-HW Nov. 20-21, 2003

Outline

Single Phase Transfer Functions:! Limitations of Existing Shape Factors! Shape Factors for Transient/Non-orthogonal Systems! Numerical Algorithm for Non-orthogonal Networks! Validation and Comparison

Two Phase Transfer Functions:! The Complete Transfer Function! Limitations of the Existing Transfer Function! New Shape Factors for Two Phase Compressible Flow! Validation, Comparison and Case Study

5SUPRI-HW Nov. 20-21, 2003

( )mmf m f

kq p pρσµ

= −

The Single Phase Transfer FunctionSi

ngle

φφ φφ T

rans

fer F

unct

ion

TF = Rate of mass transfer between matrix and fracture

V, φρ

mmf t

pq V ct

ρφ ∂= −∂

2

aL

σ =

6SUPRI-HW Nov. 20-21, 2003

Limitations of Existing Shape Factors

! Assumes pseudo-steady state (PSS)

! Only for cubic matrix blocks or orthogonalfracture systems

Sing

le φφ φφ

Tra

nsfe

r Fun

ctio

n

7SUPRI-HW Nov. 20-21, 2003

Errors due to PSS Shape Factor

Single Block (10X10 ft Matrix) DP ModelSing

le φφ φφ

Tra

nsfe

r Fun

ctio

n

L

FracturesComparison of Discrete Fracture and Dual Porosity Model for 2D Fracture

500

550

600

650

700

750

800

850

900

950

1000

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Time (days)

Ave

rage

Pre

ssur

e(p

si)

ECLIPSE Discrete DP: Lim And Aziz DP: Warren and Root DP: Kazemi

Transient PSS

8SUPRI-HW Nov. 20-21, 2003

TheTransient Shape Factor

2

2

p pDt x

∂ ∂=∂ ∂

(0, ) ( , ) ( ,0)f m mp t p p t p p x p= ∞ = =BC and IC

∞

∞

∞pm

pf

x

L

Sing

le φφ φφ

Tra

nsfe

r Fun

ctio

n

1 1 12 ( )D t P

ση

= −

1t

σ ∝

9SUPRI-HW Nov. 20-21, 2003

PSS σσσσ for Non-Orthogonal Fractures

α

L

pf

pf

pf

pfx

y2 2

2 2 2

1sin

P P PX Yτ α

∂ ∂ ∂= +∂ ∂ ∂BC and IC(0, , ) 0, (1, , ) 0, ( ,0, ) 0,( ,1, ) 0, ( , ,0) 1

P Y P Y P XP X P X Y

τ τ ττ

= = == =

22

2 2 1 sinsinLπσ α

α= +

2

2

1 sin2sin

2.5 30R

C

oασ ασ α

= ∈ =+=Sing

le φφ φφ

Tra

nsfe

r Fun

ctio

n

10SUPRI-HW Nov. 20-21, 2003

Generic Numerical Technique

( ) m mmf m f mf t

k pq p p q ct

ρσ ρφµ

∂= − = −∂

( )1 m

m f

ptD p p

σ ∂= −∂−

Sing

le φφ φφ

Tra

nsfe

r Fun

ctio

n

11SUPRI-HW Nov. 20-21, 2003

Generic Numerical Technique

( )1 m

m f

ptD p p

σ ∂= −∂−

Sing

le φφ φφ

Tra

nsfe

r Fun

ctio

n

and mm

ppt

∂∂

12SUPRI-HW Nov. 20-21, 2003

Results using Numerical TechniqueSi

ngle

φφ φφ T

rans

fer F

unct

ion

Comparison of Discrete Fracture and Dual Porosity Model for 2D Fracture

500

550

600

650

700

750

800

850

900

950

1000

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Time (days)

Ave

rage

Pre

ssur

e (p

si)

ECLIPSE Discrete DP: Lim And Aziz DP: Warren and Root DP: Kazemi DP:Variable Sigma

13SUPRI-HW Nov. 20-21, 2003

Mechanisms of 2 Phase Mass Transfer

! Pressure gradients dueto sources and sinks

! Pressure diffusion dueto compressibility

! Saturation diffusion dueto capillary forces

Px

∂∂

Py

∂∂

Two

φφ φφ Tr

ansf

er F

unct

ion

14SUPRI-HW Nov. 20-21, 2003

Complete Transfer Function

TF = Rate of mass transfer between matrix and fracture

V, φ

wS wρ

( )( )w w w w w w w w w wdm V S dS d V S V S d V dSφ ρ ρ φ ρ φ ρ φρ= + + − − +!

1 2

mf

mf mf mf

w ww w w w w

w w w

p Sq V S c Vt t

q q q

φ ρ φρ∂ ∂= − +∂ ∂

= +

Two

φφ φφ Tr

ansf

er F

unct

ion

15SUPRI-HW Nov. 20-21, 2003

Limitations of Existing Models

mf

w ww w w w w

p Sq V S c Vt t

φ ρ φρ∂ ∂= − +∂ ∂

Existing simulation models:

( )mf

rww w PD w wf

w

kq V k p pρ σµ

= −Multi φφφφ

( )mf PD f

kq V p pρ σµ

= −Single φφφφTwo

φφ φφ Tr

ansf

er F

unct

ion

16SUPRI-HW Nov. 20-21, 2003

Equations Governing FlowTw

o φφ φφ

Tran

sfer

Fun

ctio

n

( ) ( ), , , 0wc p p p p c p p p c p

pp D q S

tω λ γ ω φ ω∂ ∇ ∇ − ∇ − − = ∂

∑ "

Assumptions: Immiscible, no gravity,sources and sinks insignificant

2 p p p pp

p p

S S c pp

t tφφ

λ λ∂ ∂

∇ = +∂ ∂

Assumptions: Density and mobilityfunctions of average quantities

17SUPRI-HW Nov. 20-21, 2003

Derivation ofTw

o φφ φφ

Tran

sfer

Fun

ctio

n

2

2

w w w ww

w w

o o o oo

o o

S S c ppt t

S S c ppt t

φφλ λ

φφλ λ

∂ ∂∇ = +∂ ∂

∂ ∂∇ = +∂ ∂

111 1( ) av c

av w o w

c dPD tdS

φ φλ λ λ

−− = − +

2 ( ) w

w

SS D tt

∂∇ =∂

wSt

∂∂

2ww

S ST

∂ = ∇∂

*

0( )

t

T D dτ τ= ∫Transform:

* From Crank, 1975

18SUPRI-HW Nov. 20-21, 2003

Derivation of

Typical Imbibition Process: 1D imbibition through amatrix face, matrix initially at a constant saturation,fracture instantly filled with wetting phase.

Two

φφ φφ Tr

ansf

er F

unct

ion

2

2w wS S

T x∂ ∂=∂ ∂

max( ,0) ; (0, ) ; ( , )w wi w w wf w wiS x S S T S S S T S= = = ∞ =

( )2mfw w SD wi wq V S Sφρ σ= −"

( )wSD wi w

S S St

σ∂ = −∂

"0

( )

2 ( )tSD

D t

D dσ

τ τ=

∫"

wSt

∂∂

19SUPRI-HW Nov. 20-21, 2003

Derivation ofTw

o φφ φφ

Tran

sfer

Fun

ctio

n

2 1w w ww

w w w w

p Spt S c S c t

λφ

∂ ∂= ∇ −∂ ∂

2( ) ( )ww

p t p f tt

α∂ = ∇ +∂

wpt

∂∂

*

0( )

t

T dα τ τ= ∫

2 ( )ww

p p g TT

∂ = ∇ +∂

"

Transform:

* From Crank, 1975

20SUPRI-HW Nov. 20-21, 2003

Derivation ofTw

o φφ φφ

Tran

sfer

Fun

ctio

n

Typical PD Process: 1D fracture system, matrix initially at constant pressure,fractures suddenly reduced and maintained at a constant pressure (Lim andAziz)

2

2 ( )w wp p g TT x

∂ ∂= +∂ ∂

"(0, ) ( , )w w wfp T p L T p= =

( ,0)w wmp x p=

( ) ( )2

8( )w SDPD w wf w wi

w w

p t p p S St S c

σσ απ

∂ = − − + −∂

" 2

2PD Lπσ =

0

( )

2 ( )tSD

D t

D dσ

τ τ=

∫"

( ) ( )1 2

8mfw w w PD w wf w SD w wiq V p p V S Sρ λ σ φρ σ

π= − − −"

wpt

∂∂

21SUPRI-HW Nov. 20-21, 2003

The Complete Transfer FunctionTw

o φφ φφ

Tran

sfer

Fun

ctio

n

( ) ( )mfw w w PD w wf w SD w wiq V p p V S Sρ λσ φρ σ= − − −

21

2 2

0

8 ( ) 12 ( )

tPD SD

D t atL D d

πσ σπ τ τ

− = = + ∫

!

For the particular case of 1D parallel fractures with PSSpressure diffusion and instantaneously filled fractures, wehave:

22SUPRI-HW Nov. 20-21, 2003

The Complete Transfer FunctionTw

o φφ φφ

Tran

sfer

Fun

ctio

n

1

*

Instantly filled fracture

Gradually filling fracture

SD

mSD

atat

σσ

−

−

= ∈= ∈

2 Cubic Matrix and PSS (Lim and Aziz)

Any shape and Tran + PSS (Numeric)

( )

PD

PD

aLf t

σ

σ

= ∈

= ∈

* By comparison to results by Rangel-German

( ) ( )mfw w w PD w wf w SD w wiq V p p V S Sρ λσ φρ σ= − − −

23SUPRI-HW Nov. 20-21, 2003

ValidationTw

o φφ φφ

Tran

sfer

Fun

ctio

n

Matrix

LL

Fractures

Water Imbibing

Dimens: 200X200X200 cu.ft.

Porosity: 5%

Matrix Perm: 1 md

Fracture Perm: 10 d

Initial Pressure: 1000 psi

Fracture Pressure: 500 psi

Capillary Pressure: < 100 psi

Compressibility: 0.0001 /psi

Initial Water Saturation: 0.2

Relative Perm: Corey Type

24SUPRI-HW Nov. 20-21, 2003

Validation

Oil production rate for single porosity fine grid, using thecomplete dual porosity function and only the first term

Two

φφ φφ Tr

ansf

er F

unct

ion

25SUPRI-HW Nov. 20-21, 2003

Validation

Water imbibition rate for single porosity fine grid and usingthe complete dual porosity function

Two

φφ φφ Tr

ansf

er F

unct

ion

26SUPRI-HW Nov. 20-21, 2003

Case Study - ModelTw

o φφ φφ

Tran

sfer

Fun

ctio

n

Size: 8X8X2 Oil-Water DX = DY = 75ft DZ = 30ft

Km = 1md Kf = 10d Porm = 19% Porf = 1%

SigmaPD = 0.08 (10X10X30) Pc = 0-15 psi

Kazemi et al., 76

27SUPRI-HW Nov. 20-21, 2003

Case Study - SigmaSD

SigmaSD related to Sw by using a single blockmodel, that is, one 10x10x30 matrix block

Two

φφ φφ Tr

ansf

er F

unct

ion

SigmaSD vs Sw

0

1

2

3

4

5

6

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Sw

Sigm

aSD

(1/d

ay)

28SUPRI-HW Nov. 20-21, 2003

Case Study - Results

Additional Oil Recovery = 10%

Reduced Water Production = 15%

Two

φφ φφ Tr

ansf

er F

unct

ion

Oil and Water Production Rates

0

50

100

150

200

250

300

350

400 500 600 700 800 900 1000 1100

Time (days)

Rat

e (b

bl/d

ay)

ECLIPSE Oil Rate GPRS Oil Rate ECLIPSE Water Rate GPRS Water Rate

29SUPRI-HW Nov. 20-21, 2003

Case Study - Results

Earlier Breakthrough by 150 daysTwo

φφ φφ Tr

ansf

er F

unct

ion

Water Cut for Producer

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 200 400 600 800 1000

Time (days)

Wat

er C

ut

ECLIPSE GPRS

150 days

30SUPRI-HW Nov. 20-21, 2003

Implementation into GPRS

General Modifications:, 1

1

1 1

( ) { [ ( ) ] [ ( )]}

( ) ( ) 0

m f

nsW W n n

s p p cp p s p p cp ps p p

n n n np p cp p p c

c

pp p

F X T X W I X p p

S X S XV

t

λ ρ λ ρ

φ ρ φ ρτ

+

=

+ +

= ∆Φ + ⋅ −

−− =

∆±

∑ ∑ ∑

∑ ∑

# #

Transfer Function =mfcτ

( ) ( )mfc m PD p p cp pm pf m SD p cp pm pmip p

Vk X V X S Sτ σ λ ρ φ σ ρ = Φ − Φ − − ∑ ∑

31SUPRI-HW Nov. 20-21, 2003

mthFlowEqnModel

mthCompFlowEqnModelmthBOFlowEqnModel

mthDPCompFlowEqnModelmthDPBOFlowEqnModel

Implementation into GPRS

! Object oriented approach through inheritence andpolymorphism

! Minimum modifications to existing code! Code structured, maintain compatibility and ensure

bug-free code

32SUPRI-HW Nov. 20-21, 2003

Summary

! Existing single phase shape factors inaccurate andlimited in scope

! Numerical technique for non-orthogonal systemsand Transient+PSS flow

! Existing two phase transfer function inaccurate! New transfer function for two phase compressible

flow! Accurate modeling of fracture-matrix imbibition! New model implemented in GPRS

33SUPRI-HW Nov. 20-21, 2003

Non-orthogonal Fracture NetworksSi

ngle

φφ φφ T

rans

fer F

unct

ion

34SUPRI-HW Nov. 20-21, 2003

Non-orthogonal Fracture NetworksSi

ngle

φφ φφ T

rans

fer F

unct

ion

Pressure Response of Rhombus and Square

400

500

600

700

800

900

1000

0 0.5 1 1.5 2 2.5

Time (days)

Ave

rage

Pre

ssur

e (p

si)

Rhombic Matrix Square Matrix

Difference ~ 8%

35SUPRI-HW Nov. 20-21, 2003

Calculate ShapeFactors

Input FracturePattern

Solve for PressureUsing any Commercial

PDE Solver

Calculate AveragePressure and Derivative

Generic Numerical TechniqueSi

ngle

φφ φφ T

rans

fer F

unct

ion

36SUPRI-HW Nov. 20-21, 2003

Validation

( )mf

rww w PD w wf

w

kq V k p pρ σµ

= −

Two

φφ φφ Tr

ansf

er F

unct

ion