Recent Advances in Multiscale Finite Element …...Recent Advances in Multiscale Finite Element Methods – p.2/80 Darcy’s law and permeability Darcy’s empirical law, 1856: The

Recent Advances in Multiscale Finite Element Methods

Thomas Y. Hou

Applied Mathematics, Caltech

Collaborators: Y. Efenidev (TAMU), V. Ginting (U of Wyoming),

I. Graham (Bath), C. C. Chu (UT Austin), Xiao-Hui Wu (Exxon-Mobile)

Supported in part by NSF and DOE

Recent Advances in Multiscale Finite Element Methods – p.1/80

Introduction

• Subsurface flows and transport are affected by heterogeneities at multiple scales(pore scale, core scale, field scale).

• Because of wide range of scales direct numerical simulations are not affordable.

• Upscaling of flow and transport parameters is commonly used in practice.

• Multiscale computational methods have emerged as an effective alternative toperform upscaling for two-phase flows in strongly heterogeneous porous media.

• Multiscale methods have also been used in studying stochastic PDEs, uncertaintyquantification and other related applications.

• Most of the materials presented in this lecture can be found in the recent bookpublished by Springer:

• Y. Efendiev and T. Y. Hou, “Multiscale Finite Element Methods: Theory andApplications.” Springer, New York, 2009.


Darcy’s law and permeability

Darcy’s empirical law, 1856: The volumetric flux u(x, t) (Darcy velocity) is proportional tothe pressure gradient

u = − k

µ∇p = −K∇p, div(u) = f

where k(x) is the measured permeability of the rock, µ is the fluid viscosity, p(x) is thefluid pressure, u(x) is the Darcy velocity, f is a source or sink.

Rock Facies 1 (1−1000m)

Vugs (0.01m)Fracture (1e−4 m)

(1−1000m)Fault (1e−3m)

Rock Grains and Voids (1e−5 m)

Shale (0.01m)

Rock Facies 2


Requirements/Challenges

• Accuracy and Robustness

• Valid for different types of subsurface heterogeneity

−3

−2

−1

0

1

2

3

−1

0

1

2

3

4

5

6

7

8

9

−6

−4

−2

0

2

4

6

8

Left: two-point variogram based. Middle: synthetic channelized. Right: channelized(North Sea).

• Applicable for varying flow scenarios


Multiscale Finite Element Methods

Babuska-Osborn (83,94), Hou and Wu (1997, JCP).Consider

div(kǫ(x)∇pǫ) = f ,

where ǫ is a small parameter.

• The central idea is to incorporate the small scale information into the finite elementbases

• Basis functions are constructed by solving the leading order homogeneousequation in an element K (coarse grid or Representative Elementary Volume(RVE))

div(kǫ(x)∇φi) = 0 in K

• It is through the basis functions that we capture the local small scale information ofthe differential operator.


A simple example

0 0.2 0.4 0.6 0.8 1−0.05

0

0.05

0.1

0.15

0.2

0.25

0.3Exact solution

x

uε

−(aε(x) u’ε)’=1

0 0.2 0.4 0.6 0.8 1−0.2

0

0.2

0.4

0.6

0.8

1

1.2Basis

x

φ1 φ

2 φ

3

aǫ(x) = 1/(2 + 1.99 cos(x/ǫ)), ǫ = 0.01.



• Boundary conditions?

φi = linear function on ∂K, φi(xj) = δij

Coarse−grid Fine−grid


Basis functions

40*(2.1+sin(60*(x−y))+sin(60*y))

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

K

bilinear 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

φi = φi0 on ∂K, where φi0 are standard bilinear basis functions.



• Except for the multiscale basis functions, MsFEM is the same as the traditionalFEM (finite element method). Find phǫ ∈ V h = φi such that

k(phǫ , vh) = f(vh) ∀vh ∈ V h,

where

k(u, v) =

∫

Q

kǫij(x)∂u

∂xi

∂v

∂xjdx, f(v) =

∫

Q

fvdx

• The coupling of the small scales is through the variational formulation

• Solution has the form pǫ ≈ p0(x) + ǫNk(x/ǫ)∂

∂xkp0(x), if k(x/ǫ) is periodic.

• If standard finite element method is used (linear basis functions):

‖pǫ − phǫ ‖H1(Q) ≤ Chα‖pǫ‖H1+α(Q) = O(hα

ǫα).

Thus, h≪ ǫ, which is not affordable in practice.


Relation to other approaches

Subgrid modeling (by I. Babuska, T. Hughes, G. Allaire, T. Arbogast, and others)

Subgrid stabilization (by F. Brezzi, L Franco, J.L. Guermond, T. Hughes, A. Russo, andothers).

• Multiscale finite element methods (Hou and Wu, Allaire and Brizzi, ...)

• Multiscale finite volume methods (Jenny, Lee and Tchelepi)

• Multiscale finite element methods based on two-scale convergence (C. Schwab,...)

• Variational multiscale method and subgrid modeling (T. Hughes, F. Brezzi, T.Arbogast, G. Sangalli,...)

• A framework for adaptive multiscale method for elliptic problems (J. Nolen, G.Papanicolaou, O. Pironneau)

• Partition of Unity Methods (Babuska, Melenk, Osborn, Fish, ...)

• Heterogeneous multiscale methods (E and Engquist ...)


Brief introduction to homogenization

pǫ ∈ H10 (Q)

div(k(x,x

ǫ)∇pǫ) = f,

where k(x, y) is a periodic function with respect to y. Consider formal expansion

pǫ = p0(x, y) + ǫp1(x, y) + ǫ2p2(x, y) + ....

Taking into account

∇A(x, xǫ) = ∇xA+

1

ǫ∇yA

we have

(divx +1

ǫdivy)[k(x, y)(∇x +

1

ǫ∇y)(p0(x, y) + ǫp1(x, y) + ǫ2p2(x, y) + ...) = f.

ǫ−2 : divy(k(x, y)∇yp0(x, y)) = 0.

From the elliptic theory, we conclude that p0(x, y) = p0(x).


Brief introduction to homogenization

ǫ−1 : divy(k(x, y)∇yp1(x, y)) = −divy(k(x, y))∇xp0.

We set p1(x, y) = Nl(x, y)∂

∂xlp0, then Nl satisfies the following cell problem

divy(k(x, y)∇yNl) = −∇yikil(x, y),

with periodic boundary condition in y.

ǫ0 : divy(k(x, y)∇yp2)+divy(k(x, y)∇xp1)+divx(k(x, y)∇yp1)+divx(k(x, y)∇xp0) = f.

Taking the average and noting that 〈divyA(x, y)〉 =∫YdivyA(x, y)dy = 0, we get

divx〈k(x, y)∇yp1〉+ divx(〈k(x, y)〉∇xp0) = f.

From here, we conclude thatdivx(k

∗(x)∇xp0) = f,

where k∗(x) = 〈k(x, y) + k(x, y)∇yN〉:


Basic convergence in homogenization

For bounded domains, we have pǫ = p0(x)+ ǫN(x, y) ·∇p0 − ǫθǫ + ǫ2p2(x, y)+ ..., where

div(k∇θǫ) = 0

θǫ|∂Ω = N(x, y) · ∇p0|∂Ω.

It can be shown that (e.g., Jikov, Kozlov and Oleinik, Homogenization of differentialoperators and integral functions 1994)

‖ǫθǫ‖H1(Q) ≤ C√ǫ

and,‖pǫ − (p0(x) + ǫN(x, y) · ∇p0 − ǫθǫ)‖1,Ω ≤ Cǫ‖p0‖2,K .

See Moskow and Vogelius.


Convergence property of MsFEM

Consider kǫ(x) = k(x/ǫ), where k(y) is periodic in y.h - computational mesh size.Theorem Denote phǫ the numerical solution obtained by MsFEM, and pǫ the solution of theoriginal problem. Then,If h >> ǫ,

‖pǫ − phǫ ‖1,Q ≤ C(h+

√ǫ

h)

• This theorem shows that MsFEM converges to the correct solution as ǫ→ 0

• The ratio ǫ/h reflects two intrinsic scales. We call ǫ/h the resonance error

• The theorem shows that there is a scale resonance when h ≈ ǫ. Numericalexperiments confirm the scale resonance.


Proof of Theorem

Denote pI(x) = Ihp0(x) =∑J

j=1 p0(xj)φj(x) ∈ Vh.

Note LεpI = 0 in K and pI = Πhp0 on ∂K for any K ∈ Th. The homogenization theoryimplies that

‖pǫ − p0 − ε(p1 − θε)‖1,Ω ≤ Cε(‖f‖0,Ω + ‖p0‖2,Ω),

and‖ pI − pI0 − ε(pI1 − θIε) ‖1,K ≤ Cε(‖ f ‖0,K + | pI0 |2,K),

where pI0 is the solution of the homogenized equation on K:

L0pI0 = 0 in K, pI0 = Πhp0 on ∂K,

pI1 is given by the relation

pI1(x, y) = −χj(y)∂pI0∂xj

in K,

and θIε ∈ H1(K) is the solution of the problem:

LεθIε = 0 in K, θIε(x) = pI1(x,xε) on ∂K.


Proof of Theorem

‖ p− pI ‖1,Ω ≤ ‖ p0 − pI0 ‖1,Ω + ‖ ε(p1 − pI1) ‖1,Ω+ ‖ ε(θε − θIε) ‖1,Ω + Cε‖ f ‖0,Ω,

Now we need to estimate the terms on the right hand side.Lemma.

‖ p0 − pI0 ‖1,Ω ≤ Ch‖ f ‖0,Ω,

‖ ε(p1 − pI1) ‖1,Ω ≤ C(h+ ε)‖ f ‖0,Ω.


Proof of Theorem

Lemma.

‖ εθε ‖1,Ω ≤ C√ε‖ p0 ‖1,∞,Ω + Cε| p0 |2,Ω.

Let ζ ∈ C∞0 (R2) be the cut-off function which satisfies ζ ≡ 1 in Ω\Ωδ/2, ζ ≡ 0 in Ωδ ,

0 ≤ ζ ≤ 1 in R2, and |∇ζ| ≤ C/δ in Ω, where for any δ > 0 sufficiently small, we denoteby Ωδ as Ωδ = x ∈ Ω : dist(x, ∂Ω) ≥ δ. With this definition, it is clear that

θε − ζp1 = θε + ζ(χj∂p0/∂xj) ∈ H10 (Ω). Then (a(x

ε)∇θε,∇(θε + ζχj ∂p0

∂xj)) = 0, which

yields,

‖∇θε ‖0,Ω ≤ C√

|∂Ω| · δDδ

+ C√

|∂Ω| · δDε

+ C| p0 |2,Ω,

where D = ‖ p0 ‖1,∞,Ω and the constant C is independent of the domain Ω. We have

‖ εθε ‖0,Ω ≤ C√ε‖ p0 ‖1,∞,Ω + Cε| p0 |2,Ω.

By applying the maximum principle

‖ θε ‖0,∞,Ω ≤ xχj∂p0/∂xj0,∞, ∂Ω ≤ C‖ p0 ‖1,∞,Ω.


Proof of Theorem

By applying the above proof to each coarse grid element and summing up theircontributions, we can prove the following estimate:

Lemma. We have

‖ εθIε ‖1,Ω ≤ C( εh

)1/2‖ p0 ‖1,∞,Ω.


Resonance errors

• For problems with scale separation, we can choose h≫ ǫ in order to avoid theresonance, but for problems with continuous spectrum of scales, we cannot avoidthis resonance.

• To demonstrate the influence of the boundary condition of the basis function onthe overall accuracy of the method we perform multiscale expansion of φi

• Multiscale expansion of φi

φi = φ0(x) + ǫφ1(x, x/ǫ) + ǫθ + . . . ,

• φ1(x, x/ǫ) = Nk(x/ǫ) ∂∂xk

φ0, where Nk(x/ǫ) is a periodic function which depends

on k(x/ǫ).

• θ satisfies

div(kǫ∇θ) = 0 in K, θi = −φ1(x, x/ǫ) + (φi − φ0)/ǫ on ∂K

• Oscillations near the boundaries (in ǫ vicinity) of θi lead to the resonance error


Illustration of θ

0

4

x

θ (without oversampling)

y

θ


Oversampling technique

• To capture more accurately the small scale information of the problem, the effect ofθ needs to be moderated

• Since the boundary layer of θ is thin (O(ǫ)) we can sample in a domain with sizelarger than h+ ǫ and use only interior sampled information to construct the basisfunctions.

• Let ψk be the functions in the domain S,

div(kǫ(x)∇ψk) = 0 in S, ψk = linear function on ∂S, ψk(si) = δik.

Fine−gridCoarse−gridOversampled domain

S

K


Oversampling technique

• The base functions in a domain K ⊂ S constructed as

φi|K =∑

cijψj |K , φi(xk) = δik

• The method is non-conforming.

• The derivation of the convergence rate uses the homogenization methodcombined with the techniques of non-conforming finite element method (Efendievet al., SIAM Num. Anal. 1999)

• By a correct choice of the boundary condition of the basis functions we can reducethe effects of the boundary layer in θ.


Illustration of θ with oversampling

1.8

2.8

x

θ (with oversampling)

y

θ


Numerical Results

‖uǫ − uH‖l2 , ǫ/H = 0.64

MsFEM MsFEM-os Resolved FEMH

l2 rate l2 rate h l2

1/16 3.54e-4 7.78e-5 1/256 1.34e-41/32 3.90e-4 -0.14 3.38e-5 1.02 1/512 1.34e-41/64 4.00e-4 -0.05 1.97e-5 0.96 1/1024 1.34e-4

1/128 4.10e-4 -0.02 1.03e-5 0.95 1/2048 1.34e-4


Various generalizations

• For periodic problems or problems with scale separation, basis functions can beapproximated by

φi = φi0 + ǫχ(x/ǫ) · ∇φi0,

where φ0i is linear basis functions and χ is the periodic cell problem.

• Once basis functions are constructed, various global formulation (mixed, controlvolume finite element, DG and etc) can be used to couple the subgrid effects.

• Control volume finite element: Find pH ∈ VH such that

∫

∂Vz

k(x)∇pH · n dl =∫

Vz

q dx ∀Vz ∈ Q,

where Vz is control volume.

• Mixed multiscale finite element method has been proposed and analyzed in[Chen-Hou, Math Comptut, 2002].


MsFEM for Interface Problems with High Contrast Media

• We (Chu-Graham-Hou, Math Comp-2010) recently study the convergence property ofthe multiscale finite element method for elliptic problems with high contrast media.

−∇ · α(x)∇u = f, x ∈ Ω,

where α is piecewise constant: α(x) = α > 1 if x is inside an inclusion, and α(x) = 1

outside any inclusion. Large contrast means α≫ 1.

Γ

τ−

τ+


MsFEM for Interface Problems – Continued

• By designing an appropriate boundary condition for the multiscale bases, we provethat the MsFEM solution converges with an optimal rate independent of α:

‖u− uMSH ‖H1(Ω),α ≤ CH,

where H is the coarse grid, and C is independent of α, even as α→ ∞.

• This generalizes and improves the previous work of Z. Li, T. Lin and X. Wu [Numer.Math, 2003]. The convergence rate of their method depends on the contrast.

• Note that the convergence rate for linear FEM is much worse and the error constant Cdepends on α in an essential way.

X2

1X3 X

r+

r−

2

2

βY1

Y2 t2

n2

θ2

r+1r −1 θ1

n1

t1


MsFEM for Interface Problems – Continued

H α = 10 α = 100 α = 1000 α = 10000 α = 1000001/8 2.2721e-2 2.2770e-2 2.2720e-2 2.2850e-2 2.2908e-2

1/16 5.9381e-3 5.7494e-3 5.7563e-3 5.7796e-3 5.7805e-31/32 1.4548e-3 1.4521e-3 1.4511e-3 1.4517e-3 1.4511e-31/64 3.7294e-4 4.5074e-4 3.6735e-4 3.6177e-4 3.6343e-4rate 1.9816 1.8961 1.9840 1.9936 1.9928

Table 1: L2 norm error.H α = 10 α = 100 α = 1000 α = 10000 α = 100000

1/8 2.5490e-1 2.5826e-1 2.5081e-1 2.4871e-1 2.5382e-11/16 1.3253e-1 1.2769e-1 1.2362e-1 1.2297e-1 1.2377e-11/32 6.2156e-2 6.1687e-2 6.1456e-2 6.1289e-2 6.1355e-21/64 3.2648e-2 3.4934e-2 3.1539e-2 3.0670e-2 3.0662e-2rate 1.0009 1.0032 1.0104 1.0215 1.0402

Table 2: H1 semi-norm error.Recent Advances in Multiscale Finite Element Methods – p.28/80

Other methods that reduce resonance errors

• In Hou-Wu-Zhang (Comm. Math. Sciences-2004), we introduced aPatrov-Galerkin method in which we use multiscale basis, but linear test functions.This version of over-sampling methods can effectively remove resonance errors.

• T. Y. Hou, X.-H. Wu, and Y. Zhang, “Removing the Cell Resonance Error in theMultiscale Finite Element Method via a Petrove-Galerkin Formulation,” Comm.Math. Sciences (2004), 2(2), 185-205.

• Using a limited global information, Chen-Durlofsky (2006) andDurlofsky-Efendiev-Ginting (2007) introduced an iterative method which improvesthe accuracy of the local boundary condition for the multiscale basis via alocal-global iteration.

• Y. Chen and L. Durlofsky “Adaptive Local-Global Upscaling for General FlowScenarios in Heterogeneous Formations”, Transport in Porous Media (2006), 62,157-185.

• L. Durlofsky, Y. Efendiev, V. Ginting, An adaptive local-global multiscale finitevolume element method for two-phase flow simulations, Advances in WaterResources (2007), 30, 576-588.


Applications of MsFEM to subsurface flow simulations

Two-phase flow model. Darcy’s law for each phase

vi = −kki(Si)

µi∇pi,

i=1,2. Here k - permeability field representing the heterogeneities (micro-levelinformation), pi - the pressure, vi - velocity, ki - relative permeability, Si -saturation, µi -viscosity

phase 1 phase 2


Two-phase flow model

• p1 = p2 = p if the capillary effects are neglected. The total velocity v is given by

v = v1 + v2 = −λ(S)k∇p, λ(S) =k1(S)

µ1+k2(S)

µ2.

where S = S1, S2 = 1− S1.• Incompressibility of the total velocity implies

div(λ(S)k∇p) = 0,

• From the conservation of mass St + div(v1) = 0 we can derive

∂S

∂t+ v · ∇f(S) = 0, f(S) =

k1(S)µ1

k1(S)µ1

+k2(S)µ2


Applications of MsFEM to Two-Phase Flow

There are two ways one can apply MsFEM to solve the two-phase flow:1) Solve the pressure equation on the coarse-grid and solve the saturation equation onthe fine-grid

−div(λ(S)k∇p) = 0

∂

∂tS + v · ∇f(S) = 0,

where v = −λ(S)k∇p. Basis functions are updated only near sharp fronts.

Coarse−grid Fine−grid

2) The coarse-scale equation for the transport is obtained and coupled with MsFEM forpressure equation.


Applications of MsFEM

2) Using a perturbation approach, [Efendiev-Durlofsky-Lee, 2000] derived a macro scaleequation for the saturation equation:

∂S

∂t+ v · ∇f(S) = ∇if

′(S)2Dij∇jS

• Dij depends on two point correlation of the velocity field and S.

• The overall approach is obtained by combining the saturation equation with thepressure equation in the form div(λ(S)k∇p) = 0.

• The multiscale base functions are constructed once. The two-point correlation ofthe velocity can be found using the multiscale base functions. This approach isvery efficient and can predict the quantity of interest on a highly coarsened grid.

• Recently, [Hou-Westhead-Yang, SIAM MMS, 2006] have derived the macro-scaleequation for the saturation equation by using a systematic multiscale analysiswithout assuming smallness of the fluctuation of S.


A random porosity field with layered structure


Recovery of fine scale velocity

Left: Fine grid horizontal velocity field, N = 1024× 1024. Right: Recovered horizontalvelocity field from the coarse grid N = 64× 64 using multiscale bases.


Recovery of fine scale saturation

Left: Fine grid saturation, N = 1024× 1024. Right: Saturation computed from the recoveredfine grid velocity field.


Two-point geostatistics

fine−scale saturation plot at PVI=0.5

0

0.1

0.2

0.3

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0.5

0.6

0.7

0.8

0.9

1saturation plot at PVI=0.5 using standard MsFVEM

0

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0.9

1


Fractional flow and total flow curves

0 0.5 1 1.5 20

0.1

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0.8

0.9

1

PVI

F

finemodified MsFVMstandard MsFVM

0 0.5 1 1.5 2

3

4

5

6

7

8

9

10

11

PVIQ

finemodified MsFVMstandard MsFVM

Fractional flow and total flow for a realization of permeability field with exponential variogramand lx = 0.4, lz = 0.02, σ = 1.5.


Channelized permeability fields

Benchmark tests: SPE 10 Comparative Project

−6

−4

−2

0

2

4

6

8


Channelized reservoir

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

PVI

F

fine−scale modelstandard MsFVEM

0 0.5 1 1.5 2

50

100

150

200

250

300

350

PVIQ

fine−scale modelstandard MsFVEM

Comparison of upscaled quantities (Layer 43)



x

z

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

z

0

0.1

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0.5

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0.7

0.8

0.9

1

Comparison of saturation profile at PVI=0.5: (left) fine-scale model, (right) standard MsFVEM


MsFVEM utilizing limited global information

• The numerical tests using strongly channelized permeability fields (such as SPE 10Comparative) show that local basis functions can not accurately capture thelong-range information. There is a need to incorporate some global information.

• In [Efendiev-Ginting-Hou-Ewing, JCP 2006], we propose a modified MsFEM that useslimited global information. The main idea is to use the fine-grid solution p0 at time zeroto determine the boundary conditions for the multiscale bases.

x

x

xi

i−1

i+1

x i−1

i+1xxi

φ (

φ (

φ ()=1 )=0

)=0

i

i

i

φ ( )=0

φ ( )=0

i

x

x

i

• This approach is different from oversampling technique. Previous related work: J.Aarnes; L. Durlofsky et al.


MsFVEM utilizing global information

• If p0(xi) 6= p0(xi+1)

gi(x)|[xi,xi+1]=

p0(x)− p0(xi+1)

p0(xi)− p0(xi+1), gi(x)|[xi,xi−1]

=p0(x)− p0(xi−1)

p0(xi)− p0(xi−1).

If p0(xi) = p0(xi+1) 6= 0 then

gi|[xi,xi+1]= ψi(x) +

1

2p0(xi)(p0(x)− p0(xi+1)),

where ψi(x) is a linear function on [xi, xi+1] such that ψi(xi) = 1 and ψi(xi+1) = 0.

• The modified MsFVEM is exact for linear elliptic problem.

• When global boundary changes, then reevaluation of the basis might be needed.



0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

PVI

F

fine−scale modelmodified MsFVEMstandard MsFVEM

0 0.5 1 1.5 2

50

100

150

200

250

300

350

PVIQ

fine−scale modelmodified MsFVEMstandard MsFVEM

Comparison of upscaled quantities



x

z

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

z

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Comparison of saturation profile at PVI=0.5: (left) fine-scale model, (right) modified MsFVEM



0 0.5 1 1.5 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

PVI

F

finemodified MsFVM

0 0.5 1 1.5 20

50

100

150

200

PVI

Q

finemodified MsFVM

Comparison of upscaled quantities (Layer 43, changing boundary conditions)



fine−scale saturation plot at PVI=0.7

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1saturation plot at PVI=0.5 using modified MsFVEM

0

0.1

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0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Comparison of saturation profile at PVI=0.5: (left) fine-scale model (right) modified MsFVEM(changing boundary condition)


A Brief Analysis

• Main goal is to show that time-varying pressure is strongly influenced by the initialpressure field.

• Use the streamline-pressure coordinates:

∂ψ/∂x1 = −v2, ∂ψ/∂x2 = v1

• Set η = ψ(x, t = 0) and ζ = p(x, t = 0) and transform as follows:

x

y

− high flow channel

ζ

η − Ο(δ)

− Ο(δ)


A Brief Analysis

• The transformed pressure equation:

∂

∂η

(|k|2λ(S)∂p

∂η

)+

∂

∂ζ

(λ(S)

∂p

∂ζ

)= 0

• The transformed saturation equation:

∂S

∂t+ (v · ∇η)∂f(S)

∂η+ (v · ∇ζ)∂f(S)

∂ζ= 0

• |k|2λ(S) = |k0|2λ0(ζ, t)1Q1−δ+ |k1|2λ1(η, ζ, t)1Qδ

, λ(S) =

λ0(ζ, t)1Q1−δ+ λ1(η, ζ, t)1Qδ

.

• The pressure has the following expansion:

p(η, ζ, t) = p0(ζ, t) + δp1(η, ζ, t) + . . . ,

∂∂ζ

(λ0(ζ, t)

∂p0∂ζ

)= 0.

• Modified basis functions can exactly recover the initial pressure.


Numerical confirmation

Left: Pressure and streamline function at time t = 0.4 in Cartesian frame. Right: pressureand streamline function at time t = 0.4 in initial pressure-streamline frame.


Analysis

Assumption G. There exists a sufficiently smooth scalar valued function G(η) (G ∈ C3),such that

|p−G(psp)|1,Q ≤ Cδ,

where δ is sufficiently small.

Under Assumption G and psp ∈W 1,s(Q) (s > 2), we can prove that the multiscale finiteelement method converges with the rate given by

|p− pH |1,Q ≤ Cδ + CH1−2/s|psp|W1,s(Q) ≤ Cδ + CH1−2/s.


Extensions

Assume there exists a sufficiently smooth scalar valued function G(η), η ∈ Rn (G ∈ C3),such that

|p−G(u1, ..., un)|1,Q ≤ Cδ,

where δ is sufficiently small.Let ωi be a patch, and define φ0i to be piecewise linear basis function in patch ωi, suchthat φ0i (xj) = δij . For simplicity of notation, denote u1 = 1. Then, the multiscale finiteelement method for each patch ωi is constructed by

ψij = φ0i uj

where j = 1, .., n and i is the index of nodes. First, we note that in each K,∑ni=1 ψij = uj is the desired single-phase flow solution.

THEOREM. Assume ui ∈W 1,s(Q), s > 2, i = 1, ..., n. Then

|p− pH |1,Q ≤ Cδ + CH1−2/s.


The result of H. Owhadi and L. Zhang, CPAM, 2006

−div(k(x)∇p) = f,

where f ∈ Lp (p ≥ 2), and k(x) is rough and has some mild regularity. If k(x) is veryrough, we only expect p ∈W 1,p.

Take u1 and u2 that satisfy−div(k(x)∇ui) = 0 in Q,

ui = xi on ∂Q. Then, p(u1, u2) ∈W 2,p because it satisfies

aij∂2p

∂ui∂uj≈ f.

Owhadi and Zhang showed that the finite element method with basis functions that spanu1 and u2 converge for all f ∈ Lp (p ≥ 2).


Gridding issues

• One can improve the accuracy of multiscale simulation by choosing appropriatecoarse grids.

• Coarse grid is using some type of limited global information or local information.E.g.,

∑K−coarse block V AR(v)|K (Aarnes, Hauge, Efendiev, AWR 2007, Aarnes,

Efendiev, Jiang, Hou, 2009)

• Algorithm outline: (1) Separate the regions with different magnitude of flow; (2)Combine small blocks with a neighboring block; (3) Refine block with too muchflow; (4) Repeat Step 2;


Simulations on unstructured coarse grids

Relative Errors (layer=40, µo

µw= 3)

coarse grid frac. flow error saturation error frac. flow e rror saturation error(global) (global) (local) (local)

6 × 10 0.0144 0.0512 0.1172 0.275512 × 11 0.0093 0.0435 0.2057 0. 345912 × 22 0.0039 0.0370 0.1867 0.3158

reference

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

multiscale saturation using unstructured grid

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

multiscale saturation using structured grid


Domain decomposition for multiscale PDEs

• There have been some exciting recent developments in applying the MsFEM toDomain Decomposition for multiscale PDEs.

[1] J. Aarnes and T. Y. Hou “An Efficient Domain Decomposition Preconditioner forMultiscale Elliptic Problems with High Aspect Ratios,” Acta Mathematicae ApplicataeSinica (2002), 18 (1), 63-76.

[2] I. G. Graham, P. O. Lechner and R. Scheichl, “Domain Decomposition for MultiscalePDEs”, Numer. Math. (2007), 106, 589-626.

[3] I. G. Graham and R. Scheichl, “Robust Domain Decomposition Algorithms forMultiscale PDEs”, Numerical Methods for PDEs (2007), 23(4), 859-878.

[4] J. Galvis and Y. Efendiev, “Domain Decomposition Preconditioners for MultiscaleFlows in High-Contrast Media:, SIAM MMS (2010), 8(4), 1461-1483.


Coarse-scale approximations. DD preconditioners

div(k(x)∇p) = f.

Fine V = spanφiNf

i=1 Coarse: V0 = spanΦiNci=1 .

• Au = b, A0u0 = b0,where aij = (k∇φi,∇φj) and a0ij = (k∇Φi,∇Φj).

• Multiscale/upscaling goal: reduce ‖u− u0‖ that depends on small scales (e.g.,ǫ/H, ǫ is small scale) and the contrast (kmax/kmin).

• DD. To get to “exact” fine-scale solution, we can either (1) add more coarse basisfunctions or (2) iterate on a residual until convergence (number of iterations!).

• We use additive Schwarz preconditioner of the formB−1 = RT

0 A−10 R0 +

∑Ni=1R

Ti A

−1i Ri.

• Goal: to keep the number of iterations independent of physical parameters(cond(B−1A) ≤ C independent of contrast).


Error and its dependence on small parameters

• Besides small spatial scales (e.g., fractures, inclusions), material properties havehigh jumps (high contrast).

• Goal is to design “methods” that converge independent of physical parameters.E.g., it is well known ‖p− ph‖H1 ≤ CH‖p‖H2 ; however, ‖p‖H2 and C increasewhen spatial scales decrease and contrast increases, respectively.


Coarse spaces

• In petroleum applications, source terms are changed often. E.g., in optimizationproblems, we would like to construct a good preconditioner that can be usedmultiple times for different well configurations.


High-contrast problems

• Previous MsFEM approaches do not consider the contrast in the problem ,kmax/kmin, (except, Chu, Graham, Hou, 2009 and Berlyand and Owhadi, 2009).

• Homogenization and network results for high-contrast problems (L. Borcea and G.Papanicolaou 1997, 1998, ... Smyshlyaev, 2005-2008).

• Our objective is to construct coarse spaces (for any high-contrast heterogeneities)such that overlapping additive Schwartz preconditioners have (1)cond(B−1A) ≤ C with C independent of kmax/kmin (e.g., Graham and Scheichl,2007).


Eigenvalue problem and sparsity

• Eigenproblem div(k(x)∇ψi) = λik(x)ψi

• The weight is very important for sparsity.

• “gap” → λ1 ≤ λ2 ≤ ... ≤ λL < λL+1 ≤ ... ≤ λN ,

• λ1, ..., λL are small, asymptotically vanishing eigenvalues (∫k|∇ψi|2/

∫k|ψi|2)

• “No-separation” is possible when inclusions approach to each other (L. Borceaand G. Papanicolaou, 1997,...)


Sparse approximation

• In our work, we take the coarse space to be

V0 = Spanχiψωij = SpanΦi,

where ωi is the union of coarse grids with common vertex i (ωK is the union ofcoarse grids with common edge with K), and χi is a partition of unity function forthe node i.


Coarse spaces

• It was proved (Efendiev and Galvis, 2009) that cond(B−1A) ≤ Cλ∗

L+1if the coarse

space “spans” the eigenfunctions. A main difficulty is to prove the following fact.Theorem.

∫

K

k(v − I0v)2 1

λK,L+1

∫

ωK

k|∇v|2

∫

K

k|∇I0v|2 max1, 1

H2λK,L+1∫

ωK

k|∇v|2,

where I0v is the coarse-scale projection of v.

• From the above theorem, one can show that cond(B−1A) ≤ C(1 + (H/δ)2)

independent of contrast.


Multiscale modeling for incompressible Navier-Stokes

Motivated by the earlier work of McLaughlin-Papanicolaou-Pironneau [SIAP, 85], we(Hou-Yang-Ran, MMS 2008) develop a systematic multiscale analyis for 3D NSE.

∂tuǫ + (uǫ · ∇)uǫ = −∇pǫ + ν∆uǫ + f ,

∇ · uǫ = 0,

uǫ|t=0 = U(x) +W(x,x

ǫ).

We look for (uǫ, pǫ) of form:

uǫ = u(t,x, τ) +w(t, θ, τ,θ

ǫ), pǫ = p(t,x, τ) + q(t, θ, τ,

θ

ǫ).

Two essential new ideas:

• Reparameterize a function with infinitely many scales in frequency space anddecompose it into a formally two-scale structure.

• Use a nested multiscale expansion with a multiscale phase function, so that the initialtwo-scale structure can be preserved dynamically.


Reparameterization of multiscale initial data

Given a periodic function, we expand it in its Fourier series

u(x) =∑

k∈Z

uk exp(ikx).

We decompose the function into two parts as follows u(x) = U(x) +W(x), where U(x)

consists of low modes ≤ K/2, and

W =∑

|k|≥K/2

u(k) exp2πιk · x

=∑

k(s) 6=0,|k(l)|≤K2

u(Kk(s) + k(l)) exp2πι(Kk(s) + k(l)) · x

=∑

ks 6=0

u(s)(ks,x) exp2πιks · xǫ = W(x,

x

ǫ)

where the coefficient u(s)(k,x) only contains the Fourier modes lower than K/2, andǫ = 1/K.


A nested multiscale expansion

• Introduce a new phase function to characterize the propagation of small scales:

∂θǫ

∂t+ uǫ · ∇xθ

ǫ = 0, θǫ(x, 0) = x . (1)

• This becomes obvious when we formulate the Euler equations in the Lagrangianformulation

ω(x, t) =

(∂θǫ

∂x

)−1

ω0(θǫ(x, t),

θǫ(x, t)

ǫ).

• It is not clear what is the multiscale structure of θǫ(x, t) since its structure iscoupled to the multiscale structure of uǫ.

• Further, if we formally expand θǫ(x, t) into

θǫ(x, t) = θ(x, t) + ǫθ1(x, t,x

ǫ,t

ǫ) + · · · .

one cannot close the system. Infinitely many scales, ǫ, ǫ2, ǫ3, ..., would begenerated.


A nested multiscale expansion

• The key idea in constructing multiscale solutions for the Euler equat ion is toreformulate the problem using θǫ as a new varia ble.

• Let Xǫ be the inverse of θǫ, X be the mean of Xǫ, and θ be the mean of θǫ.• It is easy to see from the Lagrangian formulation that we can expand Xǫ as

Xǫ = X(t, α, τ) + ǫX(t, α, τ,y) ,

where X are periodic in y with zero mean, y = α/ǫ and τ = t/ǫ.

• Using the identity, θǫ = θ(t,X(t, θǫ, τ), τ), we obtain (note that α = θǫ, y = θǫ/ǫ)

θǫ = θ(t,x, τ)− ǫDxθ(t,x, τ)x(t, θ,t

ǫ,θǫ

ǫ) + · · ·

= θ(t,x, τ) + ǫθ(t,x, τ,θǫ

ǫ).

• Thus we obtain a formal two-scale expansion of θǫ. Further, we show that uǫ canbe expanded as

uǫ = u(t,x, τ) +w(t, θ, τ,θ

ǫ).


Main Results.

• The main results can be summarized as follows:• Effective equation for u:

∂tu+ (u · ∇x)u+∇x · 〈w ⊗w〉 = −∇xp+ ν∆u+ f ,

∇x · u = 0,

u|t=0 = U(x).

• Leading order small scale velocity field w with y = θ/ǫ:

∂τw +∇yw∇xθw +∇xθ⊤∇yq =

ν

ǫ∇y · (∇xθ∇xθ

⊤∇yw),

with periodic boundary conditions,(∇xθ⊤∇y

)·w = 0 and, w|τ=t=0 = W.

• Effective equation for flow map θ:

∂tθ + (u · ∇x)θ + ǫ∇x · 〈Θ⊗w〉 = 0, θ|t=0 = x.

∂tΘ+ (I +∇yΘ)∇xθw = 0,

with Θ|t=0 = 0.Recent Advances in Multiscale Finite Element Methods – p.68/80

Numerical Experiments for 3D forced NSE

DNS is performed using 5123 mesh in a periodic box to check accuracy of the multiscalemethod.

The Taylor Reynolds number is 223 in our computation (equvilent viscosity is 0.0005).

The specific form of the forcing is given by

fk =δ

N

uk√uku

∗k

(2)

N is the number of wave modes that are forced.

The above form of forcing ensures that the energy injection rate,∑f · u , is a constant

which is equal to δ. We chose δ = 0.1 for all of our runs.


3D Results

t

e k

0 5 10 15 200.2

0.4

0.6

0.8

1

1.2

1.4

Temporal evolution of total kinetic energy.


3D Results

Iso-surface plot of vorticity.


Model reduction via the Smagorinsky LES model

We first combine multiscale method with LES model to reduce computational cost. LetB = 〈w ⊗w〉, and D = 1

2(∇u+∇uT ). The well-known Smagorinsky model assumes

B =1

3kI− 2νkD (3)

where k = tr(B) is the SGS kinetic energy, and νk the eddy viscosity. Using theKolmogorov power spectra k−5/3, one derives

k = cI∆2||D||2 νk = cD∆2||D||, (4)

where ||D||2 =∑

i,j DijDij . Using the two scale model, we can also design a dynamic

model to determine the eddy viscosity. cI can be determined from

k = cI∆2||D||2 = tr(B), cD = − B⊗D

2∆2||D||3 , (5)

where B = 〈w ⊗w〉 is obtained by solving the cell equations. We solve the cell equationat 4 grid points on each dimension and average them to obtain the eddy viscosity.


Computed Eddy viscosity coefficient

0 5 10 15 20 25 30

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

Temporal evolution of computed eddy viscosity cI .


Comparison of energy spectra

100

101

102

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

Comparison of energy spectra between DNS and the averaged equation at t = 30.


Adaptive Multiscale Modeling

Recall that the multiscale model is given by

∂tu+ (u · ∇x)u+∇x · 〈w ⊗w〉 = −∇xp+ ν∆u+ f ,

∂τw +∇yw∇xθw +∇xθ⊤∇yq =

ν

ǫ∇y · (∇xθ∇xθ

⊤∇yw).

• To improve efficiency, we do not update those cells whose Reynolds stresses 〈w ⊗w〉already reach a local statistical equilibrium.

• We only update those cells that are disturbed from the equilibrium state induced by thedynamic change of the mean flow through ∇θ∇θT .

• Specifically, we only update a total P number of coarse grid cells which have thelargest difference between tn and tn−1: ‖∇θn(∇θn)T −∇θn−1(∇θn−1)T ‖.


3D Results

100

101

102

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

Comparison of energy spectrum at t = 30.0. (blue), DNS. (red), P=128; (green), P=64; (lightblue), P=32. The computations are performed for 30 eddy turn-over times.


3D Results

We first compute the distribution of the energy transfer function, ǫsgs = τ : D, betweenthe filtered scales and subgrid scales. The segment with ǫsgs < 0 corresponds to energybackscatter.

−15 −10 −5 0 5 10 150

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

DNS (solid); multiscale model with P=64 (dashed), P=32 (dasheddot).


3D Results

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

1.5

2

2.5

3

3.5

4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.5

1

1.5

2

2.5

3

3.5

4

PDF of ω · e1 (solid), ω · e2 (dashed line),ω · e3 (dotted line). DNS (left) vs LES (right), at t=20.


Conclusions

• Multiscale finite element methods provide an effective multiscale computationalmethod for flow and transport in heterogeneous porous media.

• Resonance error can be effectively removed by the over-sampling technique.

• Long range scale interaction can be captured by incorporating limited globalinformation into the multiscale bases.

• Flow-based adaptive coordinates offer a natural physical coordinate to capture thelong range interaction in a channelized media and for incompressible flows.

• Developing a systematic and robust multiscale method for engineeringapplications without scale separation remains one of the most challengingproblems. Some new ideas have already emerged. Much more work is needed.


Relevant Publications

[1] T. Y. Hou and X. H. Wu, “A Multiscale Finite Element Method for Elliptic Problems inComposite Materials and Porous Media,” J. Comput. Phys. (1997), 134, 169-189.

[2] T. Y. Hou, X. H. Wu, and Z. Cai, “Convergence of a Multiscale Finite Element Methodfor Elliptic Problems With Rapidly Oscillating Coefficients,” Math. Comput. (1999), 68,913-943.

[3] Y. R. Efendiev, T. Y. Hou, and X. H. Wu, “Convergence of A Nonconformal MultiscaleFinite Element Method,” SIAM J. Numer. Anal. (2000), 37, 888-910.

[4] Z. Chen and T. Y. Hou, “A Mixed Finite Element Method for Elliptic Problems withRapidly Oscillating Coefficients”, Math. Comput. (2002), 72(242), 541-576.[5] Y. Efendiev, T. Y. Hou, and V. Ginting, “Multiscale Finite Element Methods forNonlinear Problems” and Their Applications,” Comm. Math. Sci., Vol. 2, No. 4, pp.553-589, 2004.

[6] Y. Efendiev, V. Ginting, T. Y. Hou, and R. Ewing, “Accurate Multiscale Finite ElementMethods of Two-phase Flow Simulations,” J. Comput. Phys (2006), 220, 155-174.

[7] T. Y. Hou, D.-P. Yang, and H. Ran, “Multiscale Analysis and Computation for the 3-DIncompressible Navier-Stokes Equations”, Multiscale Modeling and Simulation (2008), 6

(4), pp. 1317-1346.

[8] Y. Efendiev and T. Y. Hou, “Multiscale Finite Element Methods: Theory andApplications.” Springer, New York, 2009.


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