1 of 37 multiscale simulation of porous media flow · 1 of 37 multiscale simulation of porous...
TRANSCRIPT
1 of 37
Multiscale simulation of porous media flowA research project funded by the Research Council of Norway
Fine scale velocity
Coarse scale velocity
↑
Geological model
Multiscalesimulation
loop
←−
Coarse scale saturation
↓
Fine scale saturation
J I
2 of 37
This work is part of the GeoScale Research Project:
Develop a numerical methodology that facilitates reservoir
simulation studies on multi-million cell geological models.
Simulations should run within a few hours on desktop computers.
J I
2 of 37
This work is part of the GeoScale Research Project:
Develop a numerical methodology that facilitates reservoir
simulation studies on multi-million cell geological models.
Simulations should run within a few hours on desktop computers.
A cornerstone in the project is a multiscale mixed finite element
method (MsMFEM) that models pressure and filtration velocity.
J I
2 of 37
This work is part of the GeoScale Research Project:
Develop a numerical methodology that facilitates reservoir
simulation studies on multi-million cell geological models.
Simulations should run within a few hours on desktop computers.
A cornerstone in the project is a multiscale mixed finite element
method (MsMFEM) that models pressure and filtration velocity.
To model the transport we explore two different strategies:
- streamline methods for convection dominated flow.
- an adaptive multiscale finite volume method.
J I
3 of 37
· Real-field geological models: 107–109 grid cells,
conventional simulators handle 105–106 grid cells.
Simulations are performed on upscaled models.
J I
3 of 37
· Real-field geological models: 107–109 grid cells,
conventional simulators handle 105–106 grid cells.
Simulations are performed on upscaled models.
· Increased demands on reservoir simulation tools:
resolution, grid cell geometries, adaptivity,....
Upscaled models can be inadequate and is a
bottle-neck in simulation workflow.
J I
3 of 37
· Real-field geological models: 107–109 grid cells,
conventional simulators handle 105–106 grid cells.
Simulations are performed on upscaled models.
· Increased demands on reservoir simulation tools:
resolution, grid cell geometries, adaptivity,....
Upscaled models can be inadequate and is a
bottle-neck in simulation workflow.
· Unlikely that the simulation gap will be closed in the foreseeable
future: multiple realizations needed to address uncertainty.
J I
4 of 37
Multiscale methods will not eliminate the need for upscaling, but ...
The ability to run simulations on geomodels is needed to validate
simulation results, and to enhance our understanding of flow
processes in reservoirs with complex structures.
Example: The oil recovery from fractured reservoirs is typically
significantly lower than the OR from non-fractured reservoirs.
J I
The Geology 5 of 37
The GeologyPorous media often have repetitive layered structures, but faults
and fractures caused by stresses in the rock disrupt flow patterns.
J I
The Geology 6 of 37
The scales that impact fluid flow in oil reservoirs range from
· the micrometer scale of pores and pore channels
· via dm–m scale of well bores and lamina sediments
· to sedimentary structures that stretch across entire reservoirs.
J I
The Geology 7 of 37
0.001 0.01 0.1 1.0 1 10 100 1000 10000
Horizontal length (m)
0.001
0.01
0.1
1
10
100
Ver
tica
l th
ickn
ess
(m
)
Laminae
Beds
Para-sequences
Geological model
Flow model
Adapted from Pickup and Hern (2002) and Barkve (2004)
Core
Probe
Log
Seismic data
J I
The Geology 8 of 37
Geological models
Geological models give a geometric reservoir description, and a
distribution of permeability k, the rocks ability to transmit fluid,
and porosity φ, the volume fraction open to flow.
0 100 200 300 400 500 6000
50
100
150
200
250
300
350
Typical porous media structures are often characterized by very
large permeability contrasts: max kmin k ∼ 103 − 1010.
J I
Simulation models 9 of 37
Simulation models
For presentational simplicity we consider a model for incompressible
and immiscible two-phase flow without gravity and capillary forces:
−∇ · k[λw(S) + λo(S)]∇p = q
φ∂tS +∇ · (fwv) = qw.
Here λi is the mobility of phase i, p pressure, S water saturation,
fw = λw/(λw + λo), and v = vw + vo the total Darcy velocity.
J I
Simulation models 10 of 37
Traditional reservoir simulation loop
Coarse geomodel
↑
Original geomodel
→
Coarse scale velocity
↓
Coarse scale saturation
J I
Simulation models 11 of 37
Reservoir simulation loop using multiscale methods
Fine scale velocity
Coarse scale velocity
↑
Geological model
←−
Coarse scale saturation
↓
Fine scale saturation
J I
Multiscale mixed finite element methods 12 of 37
Multiscale mixed finite element methodsIn a mixed FEM formulation one seeks v ∈ V and p ∈ U such that∫
Ω
k−1v · u dx−∫
Ω
p ∇ · u dx = 0 ∀u ∈ V,∫Ω
l ∇ · v dx =∫Ωql dx ∀l ∈ U.
Here V ⊂ v ∈ (L2)d : ∇ · v ∈ L2, v · n = 0 on ∂Ω and U ⊂ L2.
In MsMFEMs the approximation space for velocity V = spanψijis designed so that it embodies the impact of fine scale structures.
J I
Multiscale mixed finite element methods 13 of 37
Start with a fine grid T = T and introduce a coarsened grid
K = K with grid blocks of “arbitrary” shape.
Associate a basis function χm for pressure with each grid block:
U = spanχm : Km ∈ K where χm =
1 if x ∈ Km,
0 else.
J I
Multiscale mixed finite element methods 14 of 37
Construct a velocity basis function for each interface ∂Ki ∩ ∂Kj:
V = spanψij where ψij = −k∇φij and φij is determined by
no-flow boundary conditions on (∂Ki ∪ ∂Kj)\(∂Ki ∩ ∂Kj), and
∇ · ψij =
q(Ki) in Ki,
−q(Kj) in Kj,
where
q(K) =
|k|RK |k| if
∫Kf dx = 0,
fRK f
if∫
Kf dx 6= 0.
Homogeneous medium Heterogeneous medium
J I
Multiscale mixed finite element methods 15 of 37
Velocity basis functions
⇑
Geomodel
⇒ Coarse grid approximation space
⇓Coarse scale velocity
⇓
Fine scale velocity
The fine scale velocity field is expressed as a linear superposition of
the basis functions: v =∑
ij vijψij where the coefficients vij are
obtained from the solution of the coarse scale system.
J I
Multiscale mixed finite element methods 16 of 37
0 50 100 150 200 250 300 3500
100
200
300
400
500
600
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0 50 100 150 200 250 300 3500
100
200
300
400
500
600
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Reference MsMFEMJ I
Multiscale mixed finite element methods 17 of 37
Large scale barrier structures may not be modeled correctly,Traversing barrier Partially traversing barrier
... but the problem is easy to detect and fix automatically.
0 20 40 60 80 100 1200
20
40
60
80
100
120
−7
−6
−5
−4
−3
−2
−1
0
1
2
3
kred = 104
kyellow = 1kblue = 10−8
Fine grid = 128× 128.
J I
Multiscale mixed finite element methods 18 of 37
Grid refinement and grid adaption is straight forwardReference Uniform coarse grid Non-uniform grid Barrier grid
0 50 1000
20
40
60
80
100
120
0 50 1000
20
40
60
80
100
120
0 50 1000
20
40
60
80
100
120
0 50 1000
20
40
60
80
100
120
J I
Multiscale mixed finite element methods 19 of 37
J I
Multiscale mixed finite element methods 20 of 37
J I
Multiscale mixed finite element methods 21 of 37
050
0 20 40 60 80
100
120
140
160
180
200
220
050
0 20 40 60 80
100
120
140
160
180
200
220
050
0 20 40 60 80
100
120
140
160
180
200
220
050
0 20 40 60 80
100
120
140
160
180
200
220
MsMFEMs enjoy the following prop.:
They are accurate: flow scenarios
match closely fine grid simulations.
They are efficient: basis functions
need to be computed only once.
They are flexible: unstructured and
irregular grids are handled easily.
They are robust: suitable for mod-
eling flow in porous media with very
strong heterogeneous structures.
J I
Multiscale mixed finite element methods 22 of 37
The MsMFEM provide velocities on coarse and fine grids.
Can the MsMFEM be used as an upscaling method?
yes, but to capitalize on the enhanced resolution provided by the
MsMFEM we need to solve the saturation equation on the fine grid.
J I
Multiscale mixed finite element methods 23 of 37
4 x 4 6 x 10 10 x 22 15 x 55 30 x 1100
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8FineMsMFEMMsFVMALGUNGP−UPHA−UP
J I
Multiscale mixed finite element methods 24 of 37
4 x 4 6 x 10 10 x 22 15 x 55 30 x 1100
0.1
0.2
0.3
0.4
0.5
0.6
0.7FineMsMFEMMsFVMALGUNGP−UPHA−UP
J I
Multiscale mixed finite element methods 25 of 37
Conclusion I: Multiscale methods for elliptic equations provide a
robust and efficient tool to get accurate velocity fields on fine grids,
... but solving the saturation equation on multi-million cell
geomodels becomes a bottle-neck in large flow simulations.
Is it possible to develop a similar multiscale methodologyfor solving the saturation equation more efficiently?
J I
A multiscale method for the saturation equation 26 of 37
A multiscale method for the saturation equationFine scale velocity at tn+1 Fine scale saturation at tn
Coarse scale saturation at tn+1
↓Fine scale saturation at tn+1
J I
A multiscale method for the saturation equation 27 of 37
Assume that Sn is a saturation field on the fine grid T at t = tn,
and denote non-degenerate fine grid interfaces by γij = ∂Ti ∩ ∂Tj.
1: For each K in the coarse grid, do
Sn+1|K = Sn|K +4t∫
Kφdx
∫K
qw dx−∑
γij⊂∂K
Fij(Sn)
,where Fij(S) = maxfw(Si)vij,−fw(Sj)vij.
2: Map Sn+1|K onto the fine grid: Sn+1|K = IK(Sn+1).
J I
A multiscale method for the saturation equation 28 of 37
Here IK(S) = χK(x, t(S)), where χK is a solution of
φ∂χK
∂t+∇ · [fwv
0] = qw in KE = K ∪ T : ∂K ∩ ∂T 6= ∅,
subject to the following constraints:
Fixed velocity: v0 = v(x, t0).Initial conditions: χ0
K = S0.
Boundary conditions: fw = 1 on
γij ⊂ ∂KE : Ti ⊂ KE, vij < 0.
EK
K
To ensure that the multiscale method is mass conservative we must
choose t(S) so that∫
KIK(S)φdx = S
∫Kφdx.
J I
A multiscale method for the saturation equation 29 of 37
· Analysis (performed by Y. Efendiev) shows that the proposed
multiscale method should be accurate away from sharp fronts.
· Sharp fronts occur mainly in transient flow regions.
These regions are characterized by α ≤ S ≤ β.
· To enhance the accuracy of the multiscale method one can solve
the saturation equation on a fine grid in transient flow regions.
· Domain decomposition type localization procedures provide a
natural environment for the development of adaptive schemes.
J I
A multiscale method for the saturation equation 30 of 37
Domain decomposition method for the saturation equation:
1: For Ti ∈ KE, compute:
Sn+1/2i = Sn
i +4tφi|Ti|
∫Ti
qw(Sn+1/2) dx−∑j 6=i
F ∗ij
,
where F ∗ij =
Fij(Sn) if γij ⊂ ∂K and vij < 0.Fij(Sn+1/2) otherwise.
2: For Ti ∈ K, set Sn+1i = S
n+1/2i .
J I
A multiscale method for the saturation equation 31 of 37
Adaptive algorithm:
· Use DD method in
regions with rapid
transients (Ωfine).
· Use the multiscale
method in regions
with slow transients
(Ωcoarse).
Ω
Ω
coarse
fine
J I
A multiscale method for the saturation equation 32 of 37
J I
Benchmark: 10th SPE comparative solution project 33 of 37
Benchmark: 10th SPE comparative solution project
· Fine grid: 1.122 · 106 cells, Coarse grid: 2244 blocks.
· MsMFEM for pressure eq., MsFVM for saturation eq.
To assess solution accuracy we employ the following norms:
eF (S) =‖Sref − S‖L2
φ
‖Sref − S0ref‖L2
φ
, eC(S) =‖Sref − S‖L2
φ
‖Sref − S0ref‖L2
φ
.
Here S denotes the coarse grid saturations corresponding to S, and
‖S‖2L2
φ=
∫Ω
(Sφ)2 dx.
J I
Benchmark: 10th SPE comparative solution project 34 of 37
Water-cut curves for MsMFEM + streamline simulation:
0 500 1000 1500 20000
0.2
0.4
0.6
0.8
1
Time (days)
Wat
ercu
t
Producer A
0 500 1000 1500 20000
0.2
0.4
0.6
0.8
1
Time (days)
Wat
ercu
t
Producer B
0 500 1000 1500 20000
0.2
0.4
0.6
0.8
1
Time (days)
Wat
ercu
t
Producer C
0 500 1000 1500 20000
0.2
0.4
0.6
0.8
1
Time (days)
Wat
ercu
t
Producer D
ReferenceMMsFEM Nested Gridding
ReferenceMMsFEM Nested Gridding
ReferenceMMsFEM Nested Gridding
ReferenceMMsFEM Nested Gridding
J I
Benchmark: 10th SPE comparative solution project 35 of 37
Accuracy of saturation profiles obtained using AMsFVM:
0 0.5 10
0.05
0.1
0.15
0.2
0.25
0.3
0.35
Pore volumes injected
Fine
−grid
sat
urat
ion
erro
r
0 0.5 10
0.05
0.1
0.15
0.2
0.25
0.3
0.35
Pore volumes injected
Coa
rse−
grid
sat
urat
ion
erro
r
J I
Benchmark: 10th SPE comparative solution project 36 of 37
Water-cut curves
x: Reference solution.
–: DD method.
--: Ad. Ms. method
-·: Multiscale method
o: Coarse FV method0 0.5 1
0
0.2
0.4
0.6
0.8
1Producer 4
0 0.5 10
0.2
0.4
0.6
0.8
1Producer 2
0 0.5 10
0.2
0.4
0.6
0.8
1Producer 3
0 0.5 10
0.2
0.4
0.6
0.8
1Producer 1
J I
Conclusions and Acknowledgments 37 of 37
Conclusions and Acknowledgments
To run simulations directly on geological models require faster and
more flexible simulators than what we have available today.
Multiscale methods, as the ones presented, have a natural
flexibility, and provide a tool for running high-resolution
reservoir simulations, possibly on geological models.
Collaborators: SINTEF Texas A&M
Stein Krogstad
Vegard Kippe
Knut-Andreas Lie
Yalchin Efendiev
J I