SPE-171120 Smart Water Injection for Heavy Oil Recovery from Naturally Fractured Reservoirs
Simulation of Naturally Fractured Reservoirs with Dual ... · PDF fileSUPRI-HW Simulation of...
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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
∂∂
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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
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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
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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
∂∂
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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