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SPE 106179 Multiscale Mixed Finite Element Modeling of Coupled Wellbore / Near- Well Flow
Stein Krogstad1, Louis J. Durlofsky2
1SINTEF ICT, 2Stanford University
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SPE 106179 Multiscale Mixed Finite Element Modeling of Coupled - - PowerPoint PPT Presentation
SPE 106179 Multiscale Mixed Finite Element Modeling of Coupled - - PowerPoint PPT Presentation
SPE 106179 Multiscale Mixed Finite Element Modeling of Coupled Wellbore / Near- Well Flow Stein Krogstad 1 , Louis J. Durlofsky 2 1 SINTEF ICT, 2 Stanford University RSS07 Feb 26-28, 2007 1 Motivation: Near-well region extremely
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Outline
Motivation Multiscale Mixed Methods Drift-Flux Wellbore Flow Modeling Multiscale – Drift-Flux Coupling Numerical Experiments Conclusions / Further Work
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Multiscale Methods for Reservoir Simulation
Multiscale Finite Element Method
T. Hou, X. H. Wu, 1997
Multiscale Mixed FEM
Z. Chen, T. Hou, 2003 T. Arbogast et al., 2000 J. Aarnes et al., 2004 (group at SINTEF)
Multiscale Finite Volume Method
P. Jenny, S. H. Lee, H.A. Tchelepi, 2003+
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Standard / Multiscale Discretization
Local flow: Solve coarse equations Solve coarse equations / form fine scale flow Standard Local flow: Multiscale
Coarse model Fine model
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Mixed Finite Elements (MFEM)
= − q p v B B A
T
q p = ⋅ ∇ ∇ − = u k u λ Model equations: Weak formulation: for all Mixed discretization: ∑ ≈ ∑ ≈
j j i i
p p v ϕ ψ u Choose basis: . ) ˆ , ˆ ( V U p × ∈ u ∫ ∫ = ⋅ ∇ = ∫ ⋅ ∇ − ∫ ⋅
−
q p p p ˆ ˆ ˆ ˆ ) (
1
u u u k u λ
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Mixed / Mimetic / Multiscale
1 = ∆p
Raviart-Thomas Multiscale Mixed FEM Mimetic Finite Differences
1 − = ∆p
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Flux basis function, :
Multiscale MFEM (MsMFEM)
ψ
Initially compute basis
functions for n=1 to N
Solve coarse system
based on current saturation
Form fine scale fluxes Advance fine scale
saturation by end
p ∇ − = k ψ ∈ − ∈ = ⋅ ∇
j j i i
T x w T x w for for ψ ) (
- n
j i
T T ∂ = ⋅n ψ
t ∆
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Drift-Flux Wellbore Flow Model
d m
- V
m V C V ) ( θ + =
w w
- m
V V V α α + =
so
V
sw
V
A Ao
- /
= α
A Aw
w
/
= α 1
0 =
C ) 1 ( 53 . 1
- c
d
V V α − =
Mixture velocity (oil/water): Oil velocity:
Shi et al. (2005): Vw Vo
07 . 1 ) ( = m
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Governing Equations for Wellbore Flow
Wellbore pressure:
- so
- q
x V t = ∂ ∂ − ∂ ∂α A V q D V f g x p
m m m m m tp m
ρ ρ θ ρ 2 2 ) cos(
2
+ + = ∂ ∂
In-situ volume fraction:
θ hydrostatic friction acceleration
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Well – Reservoir Linkage
Fine grid to the annulus, well segments included in the grid Well segments treated as coarse blocks - no well model is used
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Well – Reservoir Coupling
Pressure:
Linearize wellbore
pressure equation
Couple to MsMFEM
equations
Fixed-point iteration for
initial pressure
Saturation / Holdup:
Implicit finite volume Optimal ordering of
cells
Newton iteration in
sequence for each cell / small cluster Prototype implementation: Enhancements required for full generality
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Numerical Example 1: Validation
Homogeneous permeability Compressibility (psi -1), 3x10-4 Wellbore radius (inch), 2.0 Pipe roughness (inch), 0.001 Initial saturation, 0.5 Quadratic relative perms, µo/
µw = 1
- 4200 ft
4 2 f t 600 ft
7056 fine cells 284 coarse blocks 12 well segments
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Numerical Example 1:
5200 5300 5400 5500 5600 5700 1 3 5 7 9 11 Segment number
Wellbore pressure (psia)
GPRS Fine MsMFEM
4200 4400 4600 4800 5000 5200 5400 1 3 5 7 9 11 Segment number
Wellbore pressure (psia)
GPRS Fine MsMFEM
Total rate: 1,600 STB/d Total rate: 20,000 STB/d
Pressure profiles:
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Total rate: 1,600 STB/d Total rate: 20,000 STB/d
Numerical Example 1:
0.000 0.100 0.200 0.300 0.400 0.500 0.600 1 3 5 7 9 11 Segment number In-situ oil fraction GPRS Fine MsMFEM 0.000 0.100 0.200 0.300 0.400 0.500 0.600 1 3 5 7 9 11 Segment number In-situ oil fraction GPRS Fine MsMFEM
In-situ oil fraction:
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Numerical Example 2:
Long inclined producer in heterogeneous reservoir
Two-phase oil/water
incompressible flow
Prescribed total flowrate Quadratic relative perms, µo/
µw = 10
Initially saturated with oil
600 ft 3500 ft 5 f t
85,000 fine cells 62 well segments θ between 70° and 80°
producer injector
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Numerical Example 2:
4400 4500 4600 4700 4800 4900 5000 1 11 21 31 41 51 61 Segment number Wellbore pressure (psia)
Fine MsMFEM
1000 2000 3000 4000 5000 1 11 21 31 41 51 61 Segment number Wellbore pressure (psia)
Fine MsMFEM
Pressure profiles at 0.12 PVI:
Coarse grid: 2856 blocks (factor of 30 coarsening)
Flowrate: 4,000 STB/d Flowrate: 60,000 STB/d
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Numerical Example 2:
0.2 0.4 0.6 0.8 1 1 11 21 31 41 51 61 Segment number In-situ oil fraction
Fine - low rate MsMFEM - low rate Fine - high rate MsMFEM - high rate
High flowrate: In-situ oil fraction: Low flowrate:
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Numerical Example 2:
0.2 0.4 0.6 0.8 1 1 11 21 31 41 51 61 Segment number In-situ oil fraction
Fine 17x24x21 5x5x4
4600 4650 4700 4750 4800 4850 4900 4950 1 11 21 31 41 51 61 Segment number Wellbore pressure (psia)
Fine 17x24x21 5x5x4
Coarsening factor varied from 10 to 850 Accuracy degrades with coarsening, but
physically reasonable results in all cases
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