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Title Page
Introduction
Existing Problems
Fractured Porous Media
Fractured Vuggy Media
Conclusions
11 /24
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J I
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RCOGFRby Huang
26 Oct 2010
– COMSOL 2010 ¥I«^rc¬–
Modeling two-phase flow instrongly heterogeneous
porous media
Zhaoqin Huang
Research Center for Oil & Gas Flow in Reservoir,School of Petroleum Engineering,
China University of Petroleum, Qingdao City, [email protected]://121.251.254.77/
Presented at the COMSOL Conference 2010 China
Title Page
Introduction
Existing Problems
Fractured Porous Media
Fractured Vuggy Media
Conclusions
12 /24
JJ II
J I
← → ¶
| ¢
' 4
RCOGFRby Huang
26 Oct 2010
888 ¹¹¹
1 Introduction 3
2 Fractured Porous Media 7
2.1 Discrete Fracture Model . . . . . . . . . . 7
2.2 Two-phase Flow Model . . . . . . . . . . . 9
2.3 An Example . . . . . . . . . . . . . . . . . 12
3 Fractured Vuggy Media 15
3.1 Discrete Fracture-Vug Network . . . . . . . 16
3.2 Upscaling and Equivalent Permeability . . . 17
3.3 A simple computation test . . . . . . . . . 19
4 Conclusions 21
Title Page
Introduction
Existing Problems
Fractured Porous Media
Fractured Vuggy Media
Conclusions
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JJ II
J I
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RCOGFRby Huang
26 Oct 2010
1 Introduction
Modeling Two-phase flow through strongly heterogeneousporous media is of importance in many disciplines includ-ing petroleum industry, hydrology etc.
There are, however, still some challenges in numerical sim-ulation of such flow problems especially the flows in frac-tured porous media and fractured vuggy porous media.
The aim of this report is to implement in COMSOL Mul-tiphysics a two-phase fluid flow model in strongly hetero-geneous porous media using a finite element approach.
Title Page
Introduction
Existing Problems
Fractured Porous Media
Fractured Vuggy Media
Conclusions
14 /24
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J I
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RCOGFRby Huang
26 Oct 2010
Existing Problems
• Problem 1
• Problem 2
• Problem 3
Problem 1 ... The existence of fractures or vugs can in-fluence the fluid flow in porous media largely. However areliable modeling method is difficult for fractured ( or/andvuggy) porous media due to the complex geometries offractures (or/and vugs) at multiple scales.
10cm 1mm
Title Page
Introduction
Existing Problems
Fractured Porous Media
Fractured Vuggy Media
Conclusions
15 /24
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RCOGFRby Huang
26 Oct 2010
Existing Problems
• Problem 1
• Problem 2
• Problem 3
Problem 2 ... The dual-porosity model has traditionallybeen used to simulate the flow in fractured hydrocarbonreservoirs. This approach, although very efficient, suffersfrom some important limitations.(1) One limitation is that the method cannot be appliedto disconnected fractured media and cannot represent theheterogeneity of such a system.(2) Another shortcoming is the complexity in the evalua-tion of the transfer function between the matrix and thefractures.
Title Page
Introduction
Existing Problems
Fractured Porous Media
Fractured Vuggy Media
Conclusions
16 /24
JJ II
J I
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RCOGFRby Huang
26 Oct 2010
Existing Problems
• Problem 1
• Problem 2
• Problem 3
Problem 3 ... Naturally fractured vuggy porous mediapresent multiple challenges for numerical simulations ofvarious fluid flow problems. Such media are characterizedby the presence of fractures and vugs at multiple scales.The main difficulty in numerical simulations in such mediais the co-existence of porous and free flow regions espe-cially for two-phase flow.
Title Page
Introduction
Existing Problems
Fractured Porous M . . .
Fractured Vuggy Media
Conclusions
17 /24
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RCOGFRby Huang
26 Oct 2010
2 Fractured Porous Media
2.1 Discrete Fracture Model
The discrete-fracture model describes the fractures explic-itly in the medium similarly to the single-porosity model.However, unlike the single-porosity model, the fracturesgridcells are geometrically simplified by using n−1 dimen-sional gridcells in an n-dimensional domain. As a result,computational efficiency is improved considerably.
∫Ω
FEQdΩ =∫
Ωm
FEQdΩm+∫
Ω′f
FEQdΩ′f
=∫
Ωm
FEQdΩm+a×∫
Ωf
FEQdΩf
Title Page
Introduction
Existing Problems
Fractured Porous M . . .
Fractured Vuggy Media
Conclusions
18 /24
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26 Oct 2010
Ωm(2D)
Ωf(2D)
Ωm(2D)
Ωf(1D)
Ωm(3D)
Ωf(2D)
Ωm(3D)
Ωf(3D)
original 3D model
original 2D model simplified 2D model
simplified 3D model
Title Page
Introduction
Existing Problems
Fractured Porous M . . .
Fractured Vuggy Media
Conclusions
19 /24
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RCOGFRby Huang
26 Oct 2010
2.2 Two-phase Flow Model
(1) Mathematical formulation
Mass continuity
φ∂Sα
∂t+5 vα = qα , α = w, o
Darcy’s law
vα =−K krαµα
(5pα +ραg5 z) , α = w, o
Auxiliary equationsSw+So = 1
pc (Sw) = po− pw > 0
Define the flow potential as fowllows
Φα = pα +ραgz , α = w, o
Title Page
Introduction
Existing Problems
Fractured Porous M . . .
Fractured Vuggy Media
Conclusions
110 /24
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26 Oct 2010
The whole mathematical model can be given by
0 0
0 φ
∂
∂t
Φw
Sw
+5 −
K (λw+λw) Kλop′c
Kλw 0
5Φw
Sw
=
qo+qw
qw
(2) The Galerkin finite element formulation
The formulation for water flow potential equation
∫Ω
5 [−K (λw+λo)5Φw]δΦwdΩ+∫
Ω
5 (−Kλop
′c5Sw
)δΦwdΩ
=∫
Ω
(qo+qw)δΦwdΩ
The formulation for water saturation equation
∫Ω
φ∂Sw∂t
δSwdΩ+∫
Ω
5 (−Kλw5Φw)δSwdΩ =∫
Ω
qwδSwdΩ
Title Page
Introduction
Existing Problems
Fractured Porous M . . .
Fractured Vuggy Media
Conclusions
111 /24
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26 Oct 2010
2D matrix element
1D fracture element
3D matrix element
2D fracture element
Schematic of matrix and fracture elements
13
22 2
33
4matrix element
1
2
3
4
Fracture element
2
3
=+
1
2
3
4
=
a111 a1
12 a113 0
a121 a1
22+a222
a131
0
a224
a234
a242 a2
43 a244
a123+a2
23
a132+a2
32 a133+a2
33
element 2
element 3
0 0 0 0
0 a322
0
0
0
0
0 0 0
a323
a332 a3
33
+ =
porous flow region
a111 a1
12 a113 0
a121 a1
22+a222+a3
22
a131
0
a224
a234
a242 a2
43 a244
a123+a2
23+a323
a132+a2
32+a332 a1
33+a233+a3
33
matrix element
element 1
element 2
element 1
element 2
element 1
matrix element
matrix element
element 1
element 2
element 3
element 3
element 3
FEM implementation for matrix and fracture elements
Title Page
Introduction
Existing Problems
Fractured Porous M . . .
Fractured Vuggy Media
Conclusions
112 /24
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J I
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RCOGFRby Huang
26 Oct 2010
2.3 An Example
Based on the above theory, we use the Coefficient PDEForm in COMSOL Multiphysics 3.5a to implement thediscrete fracture model.
(a) Waterflooding model (b) Permeability distribution
Title Page
Introduction
Existing Problems
Fractured Porous M . . .
Fractured Vuggy Media
Conclusions
113 /24
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26 Oct 2010
First, open the Model Navigator to set the CoefficientForm’s variables.
Now set up the flow model for the fracture is the thekey step. In this example we choose Physics −> EquationSystems −> Boundary Settings using the weak form.
Title Page
Introduction
Existing Problems
Fractured Porous M . . .
Fractured Vuggy Media
Conclusions
114 /24
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RCOGFRby Huang
26 Oct 2010
Click here to run Discrete Fracture.wmv
Fluids in fractured porous media move quickly through thefractures but also migrate, albeit relatively slowly, throughthe tiny pores within the surrounding matrix blocks.
Title Page
Introduction
Existing Problems
Fractured Porous Media
Fractured Vuggy Media
Conclusions
115 /24
JJ II
J I
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RCOGFRby Huang
26 Oct 2010
3 Fractured Vuggy Media
Fractured vuggy porous media are common in carbonaterocks, and are endemic to many of the world.s groundwa-ter aquifers and petroleum reservoirs. Recently, fracturedvuggy porous media have received much attention becausea number of fractured vuggy reservoirs have been foundworldwide that can significantly contribute to oil & gasreserves and production.
Title Page
Introduction
Existing Problems
Fractured Porous Media
Fractured Vuggy Media
Conclusions
116 /24
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J I
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RCOGFRby Huang
26 Oct 2010
3.1 Discrete Fracture-Vug Network
The fractured vuggy porous medium is a huge discretefracture-vug network space. With this concept, we pro-posed a novel model named Discrete Fracture-Vug Net-work Model (DFVN).
Horizontal Distance (m)
vug
rock matrix
fracture
Ver
tical
Dis
tanc
e (m
)
DFVN model is an extension of classic discrete fracturemodel for fractured vuggy porous media. The flow in ma-trix and fractures follows Darcy law, and the vugs are free-flow region.
Title Page
Introduction
Existing Problems
Fractured Porous Media
Fractured Vuggy Media
Conclusions
117 /24
JJ II
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RCOGFRby Huang
26 Oct 2010
3.2 Upscaling and Equivalent Permeability
Based on homogenization upscaling technique, we can de-rive the equivalent permeability.
−O2yw
js +Oyπ
js = e j , in Ys
Oy ·w js = 0, in Ys
K−1w jd+Oyπ
jd = e j , in Yd
Oy ·w jd = 0, in Yd
w js ·ns =w
jd ·ns, on Σ
2ns ·S(w j
s
)·ns = π
js −π
jd, on Σ
w js ·τs =−2
√τs ·K ·τs
αns ·S(w j
s ) ·τs, on Σ
Title Page
Introduction
Existing Problems
Fractured Porous Media
Fractured Vuggy Media
Conclusions
118 /24
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26 Oct 2010
where w jl and π
jl (l= s, d) are the Y -periodic vector fields.
The macroscopic equivalent permeability κ is then com-puted by averaging the fine-scale velocities
κi j =1|Y |
(∫Ys
(w j
s
)idy+
∫Yd
(w j
d
)idy
)∫Yd
(w j
d
)idy =
∫Ym
(w j
m
)idy+ e×
∫Yf
(w j
f
)idy , m=matrix, f=fracture
The macroscopic equivalent flux is given by the Darcy’slaw on coarse scale as ε→ 0
µ(κ)−1u+Op0 = ρf
and subject to conservation of mass
O ·u= 0
Above procedure is very similar to the one employed forupscaling the Stokes-Darcy Eqs. in a vuggy porous media.
Title Page
Introduction
Existing Problems
Fractured Porous Media
Fractured Vuggy Media
Conclusions
119 /24
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RCOGFRby Huang
26 Oct 2010
3.3 A simple computation test
0 1 2 3 4 5
0
1
2
3
4
5
a Fine-scale domain b 5×5 coarse-scale block partitioning
vug
rock matrix
fractire
Coarse-scale bolckBlock partitioning for
fine-scale domain
x1
x2
In practice, for full field-scale problems, we need to solvethe Darcy macro-model on a coarse grid, using equivalentpermeability upscaling from the fine scale Darcy-Stokescell problem. At first we should compare a fine-scale ref-erence solution with the coarse-scale model to verify thevalidity of the proposed upscaling method.
Title Page
Introduction
Existing Problems
Fractured Porous Media
Fractured Vuggy Media
Conclusions
120 /24
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26 Oct 2010
Then we can easily model the two-phase flow in suchmedia on the field-scale using the similar procedure above.
Click here to run Field-Scale Model.avi
Title Page
Introduction
Existing Problems
Fractured Porous Media
Fractured Vuggy Media
Conclusions
121 /24
JJ II
J I
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RCOGFRby Huang
26 Oct 2010
4 Conclusions
(1) Two-phase flow has been implemented withCOMSOL Multiphysics 3.5a successfully based onPressure-Saturation model, which can be conductedconveniently using Coefficient PDE Form.
(2) We coupled discrete fracture model and Pressure-Saturation model to simulate the fluid flow in fracturedmedia. The results demonstrate that the fractures arethe dominate-flow paths, and they can intensify theheterogeneity and anisotropy.
(3) The discrete fracture-vug network model provides anatural way of modeling fluids flow through fracturedvuggy porous media.
Title Page
Introduction
Existing Problems
Fractured Porous Media
Fractured Vuggy Media
Conclusions
122 /24
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RCOGFRby Huang
26 Oct 2010
Publications
1. Huang Z Q, Yao J, Li Y J, et al. Numerical Calculation ofEquivalent Permeability Tensor for Fractured Vuggy Porous MediaBased on Homogenization Theory[J]. Commun. Comput. Phys.,2011, 9(1): 180-224.
2. Huang Z Q, Yao J, Li Y J, et al. Permeability analysis offractured vuggy porous media based on homogenization theory[J].Sci China Tech Sci, 2010, 53(3): 839-847., , oæ. Äuþ!znØ¿É.0'ß5©Û.¥IÆE6: EâÆ, 2010, 40(9): 1095-1103.
3. J. Yao, Z. Q. Huang, Y. J. Li et al. Discrete Fracture-VugNetwork Model for Modeling Fluid Flow in Fractured Vuggy PorousMedia[C]. Paper SPE 130287, prepared for presentation at theCPS/SPE International Oil & Gas Conference and Exhibition inChina held in Beijing, China, 8-10 June 2010. £SPE¬ÆØ©¤
4. J. Yao, Z. Q. Huang, Y. J. Li et al. Discrete Fracture-VugNetwork Model and Its Applications to Fractured Vuggy Porous
Title Page
Introduction
Existing Problems
Fractured Porous Media
Fractured Vuggy Media
Conclusions
123 /24
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26 Oct 2010
Media[C]. prepared in the InterPore 2010 Conference and AnnualMeeting held at Texas A&M University in College Station, 14 - 17March, 2010.£InterPoreISõ0¬c¬Ø©¤
5. , , f. ¿É.hõlÑ¿Éä6ÄêÆ.[J]. hÆ, 2010, 31(4): 15-20.
6. , , ½%a. 0m6NéY0'6K[J]. ¥IhÆÆ£g,Ƥ, 2010, 34(2): 1-5.
7. , f, Ü#, . U,¿5hõlÑ¿äê[[J]. hÆ, 2010, 31(2): 91-95.
8. , , =. ÄulÑ¿.¿5hõ5Ymuê[. OÔn, ®¹^(2011).
¥IhÆ–hí'6ïÄ¥%
Title Page
Introduction
Existing Problems
Fractured Porous Media
Fractured Vuggy Media
Conclusions
124 /24
JJ II
J I
← → ¶
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RCOGFRby Huang
26 Oct 2010
Thanks for yourattention!
Questions?
