Multiscale Discretizations for Flow, Transport, and Mechanics in Porous Media

Mary F. Wheeler, Gergina Pencheva, Sunil G. Thomas

Center for Subsurface Modeling

Institute for Computational Engineering and Sciences

The University of Texas at Austin

Acknowledge Todd Arbogast, Vivette Girault, Shuyu Sun, Tim Wildey, and Ivan Yotov,

Outline

! Motivation

! Mortar mixed finite element (MMFE) methods for multiphase flow problem

! Time splitting for MMFE for multiphase flow and mixed/Godunov methods for diffusion-dispersion and reactive transport

! Numerical experiments

! Extensions to DG and DG-MMFE for flow and Galerkin for elasticity

! Conclusions

! Current and Future Work

Motivation

! Goal is to solve heterogeneous subsurface flow problems: multiphase flow, coupled with reactive transport and geomechanics in a multiscale setting.

! Applications: NAPL remediation, monitoring of nuclear wastes, C02 sequestration saline aquifers.

! Traditional method - uniform grid everywhere, too expensive. Mortars lead to attractive dynamic meshing strategies and multiphysics couplings.

! Cannot avoid if physical domain is irregular! No single smooth map to a regular computational grid exists.

Resources Recovery • Petroleum and natural gas recovery from

conventional/unconventional reservoirs • In situ mining • Hot dry rock/enhanced geothermal systems • Potable water supply • Mining hydrology

Societal Needs in Relation to Geological Systems

Site Restoration • Aquifer remediation • Acid-rock drainage

Waste Containment/Disposal• Deep waste injection • Nuclear waste disposal • CO2 sequestration • Cryogenic storage/petroleum/gas

Underground Construction• Civil infrastructure • Underground space • Secure structures

CO2 Injection and Trapping Mechanisms

Precipitated Carbonate Minerals

~800 mConfining Layer(s)

Injection Well

SupercriticalCO2

Dissolved CO2

Stratigraphic Trapping

Solubility Trapping

Mineral Trapping

Hydrodynamic Trapping

Celia et al, 2002

Diffusion-Dispersion

Solved fully-implicitly using Expanded MFEM with full-tensor

@('¤i ciw)

@t¡r ¢D¤

irciw = 0@('¤

i ciw)

@t¡r ¢D¤

irciw = 0

Ã'¤;m+1

i cm+1h;iw ¡ bTi

¢¿m+1; w

!

Ðj

+¡r ¢ zm+1

h;iw ; w¢

Ðj= 0; w 2 Wh;j

(~zm+1h;iw ;v)Ðj

= (cm+1h;iw ;r ¢ v)Ðj

¡ÐPjch;iw;v ¢ nj

®¡j

;v 2 Vh;j

(zm+1h;iw ; ~v)Ðj

= (D¤;m+1i ~zm+1

h;iw ; ~v)Ðj; ~v 2 ~Vh;i

Ã'¤;m+1

i cm+1h;iw ¡ bTi

¢¿m+1; w

!

Ðj

+¡r ¢ zm+1

h;iw ; w¢

Ðj= 0; w 2 Wh;j

(~zm+1h;iw ;v)Ðj

= (cm+1h;iw ;r ¢ v)Ðj

¡ÐPjch;iw;v ¢ nj

®¡j

;v 2 Vh;j

(zm+1h;iw ; ~v)Ðj

= (D¤;m+1i ~zm+1

h;iw ; ~v)Ðj; ~v 2 ~Vh;i

Find ~zm+1h;iw jÐj

2 ~Vh;j; zm+1h;iw jÐj

2 Vh;j; cm+1h;iw jÐj

2 Wh;j such that:Find ~zm+1h;iw jÐj

2 ~Vh;j; zm+1h;iw jÐj

2 Vh;j; cm+1h;iw jÐj

2 Wh;j such that:

Introduce ~z = ¡rc; z = D¤i~zIntroduce ~z = ¡rc; z = D¤i~z

Simplifying Assumptions

! Flow is independent of transport.! Inter-phase distribution of species assumed

to be ``locally equilibrium'' controlled, instantaneously.

! Ignore adsorption.

Preliminaries

¹Ð = [nbi=1

¹Ði : computational domain is decomposed intonon-overlapping subdomain blocks

¹Ð = [nbi=1

¹Ði : computational domain is decomposed intonon-overlapping subdomain blocks

¡ij = @Ði \ @Ðj; ¡ = [nbi;j=1¡ij; ¡i = @Ði \ ¡ = @Ðin@Сij = @Ði \ @Ðj; ¡ = [nbi;j=1¡ij; ¡i = @Ði \ ¡ = @Ðin@Ð

On each block Ði : Th;i ¡ ¯nite element partitionVh;i £Wh;i ½ H(div; Ði)£ L2(Ði)¡MFE spaces on Th;i

On each block Ði : Th;i ¡ ¯nite element partitionVh;i £Wh;i ½ H(div; Ði)£ L2(Ði)¡MFE spaces on Th;i

On each interface ¡i;j : TH;i;j ¡ interface ¯nite element gridMH;i;j ½ L2(¡i;j)¡mortar space on TH;i;j

On each interface ¡i;j : TH;i;j ¡ interface ¯nite element gridMH;i;j ½ L2(¡i;j)¡mortar space on TH;i;j

Vh =

nbM

i=1

Vh;i; Wh =

nbM

i=1

Wh;i MH =M

1·i<j·nb

MH;i;jVh =

nbM

i=1

Vh;i; Wh =

nbM

i=1

Wh;i MH =M

1·i<j·nb

MH;i;j

Mortar Domain Decomposition

Ω3

Γ13

Ω1

Γ12

Γ23

F2

F1

Ω2

Γ23

F2

F1

F3

Ω3

Ω2

Ω1F3

Γ23

F2

F1

F3

Ω3

Ω2

Ω1

Γ12

Ω3

Ω2

Ω1

Ω2

Ω3

Γ13

Ω1

Γ12 Ω2

Ω3

Γ13

Ω1

Equations for Multiphase Flow

Mass balance in each sub-domain:@('½®S®)

@t+r¢(½®u®) = q®Mass balance in each sub-domain:

@('½®S®)

@t+r¢(½®u®) = q®

Darcy's law (constitutive equation): u® = ¡kr®(S®)K

¹®(rp®¡½®grD)Darcy's law (constitutive equation): u® = ¡kr®(S®)K

¹®(rp®¡½®grD)

Saturation constraint:X

®

S® = 1Saturation constraint:X

®

S® = 1

Capillary pressure relation: pc(Sw) = pn¡pwCapillary pressure relation: pc(Sw) = pn¡pw

Continuity: On each interface ¡i;j physically meaningful BC applied:Continuity: On each interface ¡i;j physically meaningful BC applied:

p®jÐi= p®jÐj

[u® ¢ n]i;j ´ u®jÐi¢ ni + u®jÐj

¢ nj = 0p®jÐi= p®jÐj

[u® ¢ n]i;j ´ u®jÐi¢ ni + u®jÐj

¢ nj = 0

Expanded MMFE Method for Flow

Introduce a pressure gradient term to avoid inverting kr® :

~u® = ¡K=¹®(rp® ¡ ½®grD); u® = kr®(S®)~u®

Introduce a pressure gradient term to avoid inverting kr® :

~u® = ¡K=¹®(rp® ¡ ½®grD); u® = kr®(S®)~u®

In a backward Euler multi-block, we seek un®;hjÐi

2 Vh;i;

~un®;hjÐi

2 ~Vh;i; pnhjÐi

2 Wh;i; Snh jÐi

2 Wh;i; pnHj¡i;j

2 MH;i;j;Sn

Hj¡i;j2 MH;i;j for 1 · i < j · nb; such that:

In a backward Euler multi-block, we seek un®;hjÐi

2 Vh;i;

~un®;hjÐi

2 ~Vh;i; pnhjÐi

2 Wh;i; Snh jÐi

2 Wh;i; pnHj¡i;j

2 MH;i;j;Sn

Hj¡i;j2 MH;i;j for 1 · i < j · nb; such that:

!¢('½®;hS®;h)

n

¢tn; w

¶

Ði

+¡r ¢ ½n

®;hun®;h; w

¢Ði

= (qn®; w)Ði

; w 2 Wh;i

Ã!K

¹®;h

¶¡1

~un®;h;v

!

Ði

= (pn®;h;r ¢ v)Ði

¡Ðpn

®;H;v ¢ ni

®¡i

+ (½n®;hgrD;v)Ði

; v 2 Vh;i

(un®;h; ~v)Ði

= (knr®;h~u

n®;h; ~v)Ði

; ~v 2 ~Vh;iÐ £un

®;h ¢ n¤i;j

; ³®

¡i;j= 0; ³ 2 MH;i;j

!¢('½®;hS®;h)

n

¢tn; w

¶

Ði

+¡r ¢ ½n

®;hun®;h; w

¢Ði

= (qn®; w)Ði

; w 2 Wh;i

Ã!K

¹®;h

¶¡1

~un®;h;v

!

Ði

= (pn®;h;r ¢ v)Ði

¡Ðpn

®;H;v ¢ ni

®¡i

+ (½n®;hgrD;v)Ði

; v 2 Vh;i

(un®;h; ~v)Ði

= (knr®;h~u

n®;h; ~v)Ði

; ~v 2 ~Vh;iÐ £un

®;h ¢ n¤i;j

; ³®

¡i;j= 0; ³ 2 MH;i;j

Reduction to an Interface Problem

Let MH = MH £MH: De¯ne

bn(Ã; ´) =P

1·i<j·nb

P®

R¡i;j

h½n

®;hun®;h(Ã) ¢ n

i

ij´® ds;

where à = (pnw;H; Sn

w;H) 2 MH; ´ = (´w; ´w) 2 MH

Let MH = MH £MH: De¯ne

bn(Ã; ´) =P

1·i<j·nb

P®

R¡i;j

h½n

®;hun®;h(Ã) ¢ n

i

ij´® ds;

where à = (pnw;H; Sn

w;H) 2 MH; ´ = (´w; ´w) 2 MH

De¯ne the non-linear interface operator Bn : MH ! MH byDe¯ne the non-linear interface operator Bn : MH ! MH by

Then (Ã; pn®;h(Ã); Sn

®;h(Ã);un®;h(Ã)) solves the multiphase °ow

equations when Bn(Ã) = 0Then (Ã; pn

®;h(Ã); Sn®;h(Ã);un

®;h(Ã)) solves the multiphase °owequations when Bn(Ã) = 0

Interface problem is solved by \inexact Newton-GMRES" schemeInterface problem is solved by \inexact Newton-GMRES" scheme

ÐBnÃ; ´

®= bn(Ã; ´); 8´ 2 MH

ÐBnÃ; ´

®= bn(Ã; ´); 8´ 2 MH

Reactive Species Transport

Mass balance of species i in phase ® :Mass balance of species i in phase ® :

@('ci®S®)

@t+r ¢ (ci®u® ¡ 'S®Di®rci®) = r(ci®)

Di®rci® ¢ n = 0

@('ci®S®)

@t+r ¢ (ci®u® ¡ 'S®Di®rci®) = r(ci®)

Di®rci® ¢ n = 0

Di®usion-Dispersion tensor Di® = Ddi®i® + Dhyd

i® :Di®usion-Dispersion tensor Di® = Ddi®i® + Dhyd

i® :

Molecular di®usion: Dmoli® = ¿®dm;i®IMolecular di®usion: Dmoli® = ¿®dm;i®I

Physical dispersion: 'S®Dhydi® = dt;®ju®jI + (dl;® ¡ dt;®)u®u

T®

ju®jPhysical dispersion: 'S®Dhydi® = dt;®ju®jI + (dl;® ¡ dt;®)u®u

T®

ju®j

Source term: r(ci®) = rIi® + 'S®r

Ci® + qi®;

where rIi® is in°ux/e²ux from other phases,

rCi® is chemical rate of decay

qi® is a source (or sink) term

Source term: r(ci®) = rIi® + 'S®r

Ci® + qi®;

where rIi® is in°ux/e²ux from other phases,

rCi® is chemical rate of decay

qi® is a source (or sink) term

Phase-Summed Equations

Assume an equilibrium partitioning of species between phases:

ci® = !i®ci®0

Assume an equilibrium partitioning of species between phases:

ci® = !i®ci®0

Sum over all phases for a given species:Sum over all phases for a given species:

@('¤i ciw)

@t+r ¢ (ciwu¤i ¡D¤

irciw) = r¤i (cw)

Diwrciw ¢ n = 0

@('¤i ciw)

@t+r ¢ (ciwu¤i ¡D¤

irciw) = r¤i (cw)

Diwrciw ¢ n = 0

'¤i = '

P® !i®S®'¤

i = 'P

® !i®S® u¤i =P

® !i®u®u¤i =P

® !i®u®

D¤i = '

P® S®!i®Di®D¤

i = 'P

® S®!i®Di® r¤i (cw) = 'P

® rCi® ¡ riR +

P® qi®r¤i (cw) = '

P® rC

i® ¡ riR +P

® qi®

Note:X

®

rIi® + riR = 0; where riR is the in°ux/e²ux

of species i into the stationary phase

Note:X

®

rIi® + riR = 0; where riR is the in°ux/e²ux

of species i into the stationary phase

IPARS-TRCHEM Structure

TRCHEM

Advection Routines

Chemistry Routines

Dispersion/Diffusion

Flow Module

Time-splitting technique used

IPARS TRCHEM = Flow and Reactive Transport

Advection

!@'¤

i ciw

@t; w

¶

Ðj

+ (r ¢ (ciwu¤i ); w)Ðj

=

ÃX

®

qi®; w

!

Ðj

; w 2 Wj

!@'¤

i ciw

@t; w

¶

Ðj

+ (r ¢ (ciwu¤i ); w)Ðj

=

ÃX

®

qi®; w

!

Ðj

; w 2 Wj

First order Godunov scheme

Let Tmi = '¤;m

i cmh;iw; solve for T i fromLet Tm

i = '¤;mi cm

h;iw; solve for T i from

ÃT i ¡ Tm

i

¢¿m+1; w

!

Ðj

+X

E2Th;j

Ðcm;upwh;iw u

¤;m+1=2h;i ¢nE; w

®@E

=

ÃX

®

qi®; w

!

Ðj

ÃT i ¡ Tm

i

¢¿m+1; w

!

Ðj

+X

E2Th;j

Ðcm;upwh;iw u

¤;m+1=2h;i ¢nE; w

®@E

=

ÃX

®

qi®; w

!

Ðj

Solved using a Godunov scheme

Chemical Reaction

Solved by explict ODE integration using Runge-Kutta

De¯ne ©(t) ´ diagf'¤i (t)g; T = T(t) ´ ©(t)cw; andDe¯ne ©(t) ´ diagf'¤i (t)g; T = T(t) ´ ©(t)cw; and

r¤;Ci (T) ´ 'X

®

rCi®(©¡1(t)T) Then

@Ti

@t= r¤;Ci (T)r¤;Ci (T) ´ '

X

®

rCi®(©¡1(t)T) Then

@Ti

@t= r¤;Ci (T)

k1;i = ¢¿m+1 r¤;Ci (T)

k2;i = ¢¿m+1 r¤;Ci

!T +

1

2k1

¶

bTi = T + k2;i

k1;i = ¢¿m+1 r¤;Ci (T)

k2;i = ¢¿m+1 r¤;Ci

!T +

1

2k1

¶

bTi = T + k2;i

Second order Runge-Kutta scheme

Accounting for Non-matching grids in Transport Problem

Pj : L2(¡j) ! L2(¡k) is an L2-orthogonal proj. s.t. 8Á 2 L2(¡j)

hÁ¡ PjÁ;v ¢ nji¡k;j= 0; 8v 2 Vh;i; 8k such that Ðk \ Ðj 6= ;

Pj : L2(¡j) ! L2(¡k) is an L2-orthogonal proj. s.t. 8Á 2 L2(¡j)

hÁ¡ PjÁ;v ¢ nji¡k;j= 0; 8v 2 Vh;i; 8k such that Ðk \ Ðj 6= ;

Algorithm

NAPL Remediation

! Bio-remediation of NAPL using microbes

! Advection-Diffusion-Reaction! Discontinuous permeability

field with barriers! Two flowing phases - quarter-

five spot! External BC: no-flow and

zero diffusive flux! IC: NAPL, microbes occupy

0 < y < 40 ft and O2, N2occupy 40 < y < 400 ft.

! Domain: 20 ft x 400 ft x 400 ft! Reference case: NX=20,

NY=40, NZ=40

Flow Pattern in Multi-block

Reference Solution

Concentrations of tracer, NAPL, microbes and bio-degraded product at 100 days

Comparison to Mortar Scheme

Tracer & NAPL concentrations at 5, 50, 100 days

Comparison to Mortar Scheme, cont’d

Microbe & product concentrations at 5, 50, 100 days

Multinumeric Extensions

! DG and Mixed FEM can be combined for treating flow using mortar spaces

! DG is applicable for both flow and transport on non-matching grids

! Examples for single-phase slightly compressible flow follow

DG-MFEM, 3 blocks with a fault

! 250 x 100 x 100 ft! 2 ft wide fault: 10000

mD, !=0.01! 4 geological layers: 10,

100, 300, 10 mD, !=0.2! BC:

500 psi at x=0400 psi at x=250,noflow o.w.

! r=2, k=0 (RT0), m=0

3 blocks with a fault : Solution

DG-DG, Oxbow problem

! 4 blocks with 6 wells! 900 x 500 x 24 ft! 1 production, 5

injection wells! Kxx = Kyy=200,30,40,

Kzz=25,5,3,!=0.22,0.08,0.09

! BC: Pinj = 700 psi, Pprod = 500 psi, noflow on the outer bdry

! nonmatching grids, r=2, m=1

Unstructured Mesh

Top view, 4 blocks Magnified grid around well

Linear Elasticity Problem

¡r ¢ ¾(u) = f in Ð

u = uD on ¡D

¾(u)n = tN on ¡N

¡r ¢ ¾(u) = f in Ð

u = uD on ¡D

¾(u)n = tN on ¡N

¾(u) = ¸r ¢ uI + 2¹"""(u) - anisotropic Hooke's law

"""(u) =1

2

¡ru +ruT

¢- kinematic equation

¾(u) = ¸r ¢ uI + 2¹"""(u) - anisotropic Hooke's law

"""(u) =1

2

¡ru +ruT

¢- kinematic equation

u : displacement

""" : linearized strain tensor

¾ : Cauchy stress

¸ > 0; ¹ > 0 : Lam¶e parameters

u : displacement

""" : linearized strain tensor

¾ : Cauchy stress

¸ > 0; ¹ > 0 : Lam¶e parameters

Find u 2 H1(Ð) s.t.Find u 2 H1(Ð) s.t.

Domain Decomposition

Ð = Ð1 [ Ð2 [ ¡12Ð = Ð1 [ Ð2 [ ¡12

On each block Ði :

¡r ¢ ¾(u) = f in Ði

u = uD on ¡Di= @Ði \ ¡D

¾(u)n = tN on ¡Ni= @Ði \ ¡N

On each block Ði :

¡r ¢ ¾(u) = f in Ði

u = uD on ¡Di= @Ði \ ¡D

¾(u)n = tN on ¡Ni= @Ði \ ¡N

Find u with ujÐi2 H1(Ði) s.t.Find u with ujÐi2 H1(Ði) s.t.

On the interface ¡12 :

[u] = 0; [¾(u)] n1 = 0; on ¡12 : transmission conditions

On the interface ¡12 :

[u] = 0; [¾(u)] n1 = 0; on ¡12 : transmission conditions

Discrete Spaces

Th(Ði)¡ a conforming partition of Ði; i = 1; 2

EH ¡mortar ¯nite element partition on ¡12; independent of

interior meshes

Th(Ði)¡ a conforming partition of Ði; i = 1; 2

EH ¡mortar ¯nite element partition on ¡12; independent of

interior meshes

Xh;i = fvh;i 2 L2(Ði) : vh;ijeÐi2 H1(eÐi) ; 8E 2 Th;i; vh;ijE 2 IP k(E)g

¤H = f¸H 2 L2(¡12) : 8¿ 2 EH; ¸Hj¿ 2 IP l(E)gXh;i = fvh;i 2 L2(Ði) : vh;ijeÐi

2 H1(eÐi) ; 8E 2 Th;i; vh;ijE 2 IP k(E)g¤H = f¸H 2 L2(¡12) : 8¿ 2 EH; ¸Hj¿ 2 IP l(E)g

8vh 2 Xh ; jvhjXh=

¡ 2X

i=1

jvh;ij2h;i

¢1=28vh 2 Xh ; jvhjXh=

¡ 2X

i=1

jvh;ij2h;i

¢1=2

jvh;ij2h;i = ¸¡kdiv vh;ik2

L2(eÐi)+

X

E2Oi

kdiv vh;ik2L2(E)

¢

+ 2 ¹¡k"(vh;i)k2

L2(eÐi)+

X

E2Oi

k"(vh;i)k2L2(E)

¢

jvh;ij2h;i = ¸¡kdiv vh;ik2

L2(eÐi)+

X

E2Oi

kdiv vh;ik2L2(E)

¢

+ 2 ¹¡k"(vh;i)k2

L2(eÐi)+

X

E2Oi

k"(vh;i)k2L2(E)

¢

Mortar Variational Form2X

i=1

³Z

Ð i

¾ (u i) : ²(v i) ¡Z

¡Di

¡¾ (u i)nÐ ¢ v i + sD¾ (v i)nÐ ¢ u i

¢

¡Z

¡12

¡¾ (u i)n i ¢ v i ¡ ¾ (v i)n i ¢ u i

¢

+ (¸ + 2 ¹)¡ X

°2¡h;i

¾°

h°

Z

°

u i ¢ v i +X

¿2EH

¾¿

H¿

Z

¿

u i ¢ v i

¢´

=

Z

Ð

f ¢ v +

Z

¡N

tN ¢ v

¡2X

i=1

³sD

Z

¡Di

¾ (v i)nÐ ¢ uD ¡ (¸ + 2 ¹)X

°2¡h;i

¾°

h°

Z

°

uD ¢ v i

´

+

2X

i=1

³Z

¡12

¾ (v i)n i ¢ ¸ + (¸ + 2 ¹)X

¿ 2EH

¾¿

H¿

Z

¿

¸ ¢ v i

´; 8v

2X

i=1

³Z

Ð i

¾ (u i) : ²(v i) ¡Z

¡Di

¡¾ (u i)nÐ ¢ v i + sD¾ (v i)nÐ ¢ u i

¢

¡Z

¡12

¡¾ (u i)n i ¢ v i ¡ ¾ (v i)n i ¢ u i

¢

+ (¸ + 2 ¹)¡ X

°2¡h;i

¾°

h°

Z

°

u i ¢ v i +X

¿2EH

¾¿

H¿

Z

¿

u i ¢ v i

¢´

=

Z

Ð

f ¢ v +

Z

¡N

tN ¢ v

¡2X

i=1

³sD

Z

¡Di

¾ (v i)nÐ ¢ uD ¡ (¸ + 2 ¹)X

°2¡h;i

¾°

h°

Z

°

uD ¢ v i

´

+

2X

i=1

³Z

¡12

¾ (v i)n i ¢ ¸ + (¸ + 2 ¹)X

¿ 2EH

¾¿

H¿

Z

¿

¸ ¢ v i

´; 8v

Z

¡12

[¾(u)]12n12 ¢ ¹¡ (¸ + 2 ¹)X

¿2EH

¾¿

H¿

2X

i=1

Z

¿

(ui ¡ ¸) ¢ ¹ = 0; 8¹ 2 H12(¡12)

Z

¡12

[¾(u)]12n12 ¢ ¹¡ (¸ + 2 ¹)X

¿2EH

¾¿

H¿

2X

i=1

Z

¿

(ui ¡ ¸) ¢ ¹ = 0; 8¹ 2 H12(¡12)

Error Estimates

juh ¡ ujXh+

³(¸ + 2 ¹)

2X

i=1

³ X

°2¡h;i

¾°

h°kuh;i ¡ uDk2

L2(°)

+X

°2Sh;i

¾°

h°k[uh;i]°k2

L2(°) +X

¿2EH

¾¿

H¿kuh;i ¡ ¸Hk2

L2(¿ )

´´1=2

· C³hr¡1

³³ h

H

´1=2

+³H

h

´1=2´+ H ¹r¡1=2Z

´;

juh ¡ ujXh+

³(¸ + 2 ¹)

2X

i=1

³ X

°2¡h;i

¾°

h°kuh;i ¡ uDk2

L2(°)

+X

°2Sh;i

¾°

h°k[uh;i]°k2

L2(°) +X

¿2EH

¾¿

H¿kuh;i ¡ ¸Hk2

L2(¿ )

´´1=2

· C³hr¡1

³³ h

H

´1=2

+³H

h

´1=2´+ H ¹r¡1=2Z

´;

where r = min(k + 1; s) and ¹r = min(` + 1; s¡ 1=2)where r = min(k + 1; s) and ¹r = min(` + 1; s¡ 1=2)

Z = 1; if DG strips are usedZ = 1; if DG strips are used

Z =³H

h

´1=2

; if CG everywhereZ =³H

h

´1=2

; if CG everywhere

Elastic Beam

¸ = Eº(1+º)(1¡2º); ¹ = E

2(1+º)¸ = Eº(1+º)(1¡2º); ¹ = E

2(1+º)

u =

!¡3®x2y

®x3 + 3®¸¸+2¹ x(y2 ¡ c2)

¶u =

!¡3®x2y

®x3 + 3®¸¸+2¹ x(y2 ¡ c2)

¶

f =

Ã6®¹3¸+4¹

¸+2¹ y

0

!

f =

Ã6®¹3¸+4¹

¸+2¹ y

0

!

º = 0:3º = 0:3

E = 2:0E+5 N m¡2E = 2:0E+5 N m¡2

L = 10 mL = 10 m

® = ¡1:0E-3 m¡2® = ¡1:0E-3 m¡2

c = 0:5mc = 0:5m

Meshes and Solution

Initial subdomain & mortar mesh Deformed mesh

Displacement in x-direction

Displacement in y-direction

Convergence Rates

Linear

Subdomain

Approx.

(k=1)

Quadratic

Subdomain

Approx.

(k=2)

Conclusions

! Mortar methods defined and implemented for " Fully implicit multiscale method (MMFE) for multiphase flow

coupled to a mixed/Godunov method for advection-diffusion-reaction problems on non-matching grids.

" Elasticity

! Variably refined sub-domains results in significant savings in computational time (1 domain with fine grid takes twice the time as 3 domains) for multphase flow coupled to reactive transport" Multiblock domain solution agrees very well with single-domain

fine-everywhere

! Mortar approach allows for legacy code reuse and " Cheaper way to handle irregular geometries" Ideally suited for handling geological faults, fractures, etc.

Current and Future Work

! Explore dynamic load balancing for treating reactions in parallel computations

! Apply error estimates for flow & transport to make suitable choice of sub-domain grids and mortar degrees of freedom

! Ading sharper a posteriori error estimators for adaptive mesh refinement (with M. Vohralík)

! Geomechanics model extensions to include permeability dependence on stress and coupling to nonisothermal EOS compositional flow model

References

[1] Vivette Girault, Shuyu Sun, Mary F. Wheeler, and Ivan YotovCoupling Discontinuous Galerkin and Mixed Finite Element Discretizations using Mortar Finite Elements, SIAM Journal on Numerical Analysis, 46m pp.949-979 (2208)

[2] Todd Arbogast, Gergina Pencheva, Mary F. Wheeler, and Ivan YotovA Multiscale Mortar Mixed Finite Element MethodMultiscale Modeling & Simulation, 6:319--346, 2007.

[3] Todd Arbogast, Lawrence C. Cowsar, Mary F. Wheeler, and Ivan YotovMixed finite element methods on non-matching multiblock gridsSIAM Journal on Numerical Analysis, 37:1295--1315, 2000.

[4] Gergina Pencheva, Sunil Thomas, and Mary F. Wheeler, Mortar Coupling of Mulltiphase Flow and Reactive Transport on Non-matching Grids, Finite Volumes for Complex Applications V, (2008).

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