1 of 37 multiscale simulation of porous media flow · 1 of 37 multiscale simulation of porous...

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