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Multipoint Flux Mixed Finite Element Method

in Porous Media Applications

Part I: Introduction and Multiscale Mortar Extension

Guangri Xue (Gary)

KAUST GRP Research FellowCenter for Subsurface Modeling

Institute for Computational Engineering and SciencesThe University of Texas at Austin

In collaboration with:Mary F. Wheeler, The University of Texas at AustinIvan Yotov, University of Pittsburgh

Acknowledgement:

GRP Research Fellowship, made by KAUST

KAUST WEP Workshop, Saudi Arabia, 1/30/2010

Center for Subsurface ModelingInstitute for Computational Engineering and Sciences

The University of Texas at Austin, USA

Modeling Carbon Sequestration

CO2 Sequestration Modeling

Key Processes

CO2/brine mass transfer Multiphase flow

During injection (pressure driven) After injection (gravity driven)

Geochemical reactions Geomechanical modeling

Numerical simulations

Characterization (fault, fractures) Appropriate gridding Compositional EOS Parallel computing capability

Key Processes

CO2/brine mass transfer Multiphase flow

During injection (pressure driven) After injection (gravity driven)

Geochemical reactions Geomechanical modeling

Numerical Simulations

Characterization (fault, fractures) Appropriate gridding Compositional EOS Parallel computing capability

Center for Subsurface ModelingInstitute for Computational Engineering and Sciences

The University of Texas at Austin, USA

Corner Point Geometry

General hexahedral grid (with non-planar faces) Fractures and faults Pinch-out Layers Non-matching

Center for Subsurface ModelingInstitute for Computational Engineering and Sciences

The University of Texas at Austin, USA

Outline

Some locally conservative H(div) conforming method Multipoint flux mixed finite element method (MFMFE) Multiscale Mortar MFMFE Numerical examples Summary and Conclusions

Center for Subsurface ModelingInstitute for Computational Engineering and Sciences

The University of Texas at Austin, USA

Some locally conservative H(div) conforming method

Mixed Finite ElementRaviart, Thomas 1977; Nedelec 1980; Brezzi, Douglas, Marini 1985;

Brezzi, Douglas, Duran, Fortin 1987; Brezzi, Douglas, Duran, Marini

1985; Chen, Douglas 1989, Shen 1994; Kuznetsov, Repin 2003;

Arnold, Boffi, Falk 2005; Sbout, Jaffre, Roberts 2009...

Mimetic Finite DifferenceShashkov, Berndt, Hall, Hyman, Lipnikov, Morel, Moulton, Roberts,

Steinberg, Wheeler, Yotov ...

Cell-Centered Finite DifferenceRussell, Wheeler 1983; Arbogast, Wheeler, Yotov 1997; Arbogast,

Dawson, Keenan, Wheeler, Yotov 1998 ...

Multipoint Flux ApproximationAavatsmark, Barkve, Mannseth 1998; Aavatsmark 2002; Edwards

2002; Edwards, Rogers 1998, ...

Multipoint Flux MFEWheeler, Yotov 2006; Ingram, Wheeler, Yotov 2009; Wheeler, X.,

Yotov 2009, 2010

Center for Subsurface ModelingInstitute for Computational Engineering and Sciences

The University of Texas at Austin, USA

Multipoint Flux Mixed Finite Element (MFMFE)1

Find u H(div), p L2,

(K1u,v) (p, v) = 0, v H(div)( u, q) = (f, q), q L2

MFMFE method: find uh Vh, ph Wh,

(K1uh,v)Q (p, v) = 0, v Vh( u, q) = (f, q), q Wh

Finite element space: Vh(E) and Wh(E)

Vh(E) ={Pv|v V (E)

}, Wh(E) =

{q|q W (E)

}Numerical quadrature rule:

(K1uh,vh)Q =ETh

(K1uh,vh)Q,E =ETh

(1

JBTK1Buh, vh

)Q,E

Center for Subsurface ModelingInstitute for Computational Engineering and Sciences

The University of Texas at Austin, USA

Multipoint Flux Mixed Finite Element (MFMFE)2

FEM space on E:

Simplicial element [Brezzi, Douglas, Marini 1985; Brezzi, Douglas, Duran, Fortin 1987]:

V(E) = P1(E)d, W (E) = P0(E),

2D square [Brezzi, Douglas, Marini 1985]:

V (E) = BDM1(E) =

(1x+ 1y + r1 + rx

2 + 2sxy2x+ 2y + r2 2rxy sy2

)W (E) = P0(E)

3D cube [Ingram, Wheeler, Yotov 2009]:

V (E) = BDDF1(E) + r2curl(0,0, x2z)T + r3curl(0,0, x

2yz)T

+ s2curl(xy2,0,0)T + s3curl(xy

2z,0,0)T

+ t2curl(0, yz2,0)T + t3curl(0, xyz

2,0)T

W (E) = P0(E)

Center for Subsurface ModelingInstitute for Computational Engineering and Sciences

The University of Texas at Austin, USA

Multipoint Flux Mixed Finite Element (MFMFE)3

Numerical quadrature rule on V (E):

(K1uh,vh)Q,E =(

1

JBTK1Buh, vh

)Q,E

Symmetric [Wheeler and Yotov 2006]:

(1

JBTK1Buh, vh

)Q,E

=|E|nv

nvi=1

(1

JBTK1Buh vh

)|ri

nv: number of vertices of E.

Center for Subsurface ModelingInstitute for Computational Engineering and Sciences

The University of Texas at Austin, USA

Properties of MFMFE1

(M NT

N 0

)(UP

)=

(0F

)Basis functions in V (E):

v11(r1) n1 = 1, v11(r1) n2 = 0v11(ri) nj = 0, for i 6= 1, j = 1,2

(1

JBTK1Bv11, v11

)Q,E6= 0(

1

JBTK1Bv11, v12

)Q,E6= 0(

1

JBTK1Bv11, vij

)Q,E

= 0, i 6= 1

M is block diagonal. Cell-centered scheme:

NM1NTP = F

Center for Subsurface ModelingInstitute for Computational Engineering and Sciences

The University of Texas at Austin, USA

Properties of MFMFE2

Locally conservative Cell-centered scheme, solver friendly Equivalent to multipoint flux approximation method Accurate for full tensor coefficient, simplicial grids, h2-quadrilateral

grid, and h2-hexahedral grid with non-planar faces

Superconvergent

Center for Subsurface ModelingInstitute for Computational Engineering and Sciences

The University of Texas at Austin, USA

Convergence Results of MFMFE

Symmetric method

Theorem [Wheeler and Yotov 2006, Ingram, Wheeler, and Yotov 2009] On simplicial

grids, h2-parallelograms, and h2-parallelepipeds

u uh+ div(u uh)+ p ph ChQhp ph Ch2, for regular h2-parallelpipeds

Proposition On h2-parallelogram and K-orthogonal grids,

RuRuh Ch2

R: RT 0 projection

Open question for non-orthogonal grid.

Center for Subsurface ModelingInstitute for Computational Engineering and Sciences

The University of Texas at Austin, USA

Multiscale Mortar MFMFE

Center for Subsurface ModelingInstitute for Computational Engineering and Sciences

The University of Texas at Austin, USA

Multidomain variational formulation

Vi = H(div; i), V =ni=1

Vi,

Wi = L2(i), W =

ni=1

Wi = L2().

i,j = H1/2(i,j), =

1i

Multiscale Mortar MFMFE: formulation

Multiscale Mortar: Mixed Finite Element

Theorem [Arbogast, Pencheva, Wheeler & Yotov 2007]:

Vh =n

i=1 Vh,i, Wh =n

i=1 Wh,i, MH =n

i=1 MH,i,j

i j

ijVh,i: RT, BDM, .., spacesWh,j : piecewise polynomial

MH,i,j : piecewise polynomial

Find uh Vh, p Wh, and MH , for i = 1, , n,(K1uh,v)i (ph, v)i = < H ,v ni >i v Vh,i( uh, q)i = (f, q) q Wh,in

i=1 < uh ni, >i= 0 MH

u uh = O(Hm+1/2 + hk+1)p ph = O(Hm+3/2 + hk+1)

m: degree of mortar approximation polynomial space MHk: order of approximation for velocity and pressure

Vh =ni=1

Vh,i, Wh =ni=1

Wh,i

H =

1i

Multiscale Mortar MFMFE: an interface formulation1

Interface problem:

dH(H , ) = gH(), H ,

dH : L2() L2() R for , L2() by

dH(, ) =ni=1

dH,i(, ) = ni=1

Ruh() ni, i.

gH : L2() R:

gH() =ni=1

gH,i() =ni=1

Ruh ni, i,

Star problem: (uh(), ph()) Vh Wh solve, for 1 i n,

(K1uh(),v)Q,i (ph(), v)i = ,Rv nii, v Vh,i,

( uh(), w)i = 0, w Wh,i.

Center for Subsurface ModelingInstitute for Computational Engineering and Sciences

The University of Texas at Austin, USA

Multiscale Mortar MFMFE: an interface formulation2

Bar problem: (uh, ph) Vh Wh solve, for 1 i n,

(K1uh(),v)Q,i (ph(), v)i = g,Rv nii/ i, v Vh,i,( uh(), w)i = 0, w Wh,i.

with

uh = uh(H) + uh, ph = p

h(H) + ph.

Center for Subsurface ModelingInstitute for Computational Engineering and Sciences

The University of Texas at Austin, USA

Weakly Continuous Velocity Sapce

Vh,0 =

v Vh :ni=1

Rv|i ni, i = 0 H

.Assumption: For any H,

0,i,j C(QRh,i0,i,j + Q

Rh,j0,i,j

), 1 i < j n. (1)

Lemma 1 Under assumption (1), there exists a projection operator

0 :(H1/2+()

)dV Vh,0 such that

( (0q q), w) = 0, w Wh,

0qq .ni=1

qr+1/2,ihr(h1/2 +H1/2), 0 r 1,

0q q .ni=1

q1,ih1/2(h1/2 +H1/2).

Center for Subsurface ModelingInstitute for Computational Engineering and Sciences

The University of Texas at Austin, USA

Solvability of Multiscale Mortar MFMFE method

(K1uh,v)Q,i (ph, v)i = g,Rv nii/ H ,Rv nii, v Vh,i, (2)

( uh, w)i = (f, w)i, w Wh,i, (3)ni=1

Ruh ni, i = 0, H . (4)

Lemma 2 Assume that (1) holds. Then, there exists a unique solutionof (2)-(4).Sketch of Proof:

1. Let f = 0 and g = 0, v = uh, w = ph, and = H,

ni=1

(K1uh,uh)Q,i = 0, thus uh = 0.

2. q H1() s.t. q = ph Taking v = 0q in (2),