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Global Jacobian Mortar Algorithms for Multiphase Flow in Porous Media Ben Ganis* Collaborators: Kundan Kumar*, Gergina Pencheva*, Mary F. Wheeler*, Ivan Yotov** *Center for Subsurface Modeling, ICES, UT Austin **University of Pittsburgh ICES

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SLIDE 1

Center for Subsurface Modeling

Ben Ganis | GJ Mortar Algorithms | bganis@ices.utexas.edu

Global Jacobian Mortar Algorithms for Multiphase Flow in Porous Media

ICES Seminar-Babuska Forum Series November 7, 2014

Ben Ganis* Collaborators: Kundan Kumar*, Gergina Pencheva*, Mary F. Wheeler*, Ivan Yotov**

*Center for Subsurface Modeling, ICES, UT Austin **University of Pittsburgh

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SLIDE 2

Center for Subsurface Modeling

Ben Ganis | GJ Mortar Algorithms | bganis@ices.utexas.edu

Multiscale Mortar Mixed FEM

  • Mortar finite elements are a domain decomposition

technique to couple unknowns across:

– Multiple Scales – Multiple Physics – Multiple Numerics – Multiple Processors

  • Note that Domain Decomposition is not the same as

“Data Decomposition”.

  • The “Global Jacobian” algorithms developed in this

research seek to have the best of both worlds.

Interfaces Γkl Subdomains Ωk

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SLIDE 3

Center for Subsurface Modeling

Ben Ganis | GJ Mortar Algorithms | bganis@ices.utexas.edu

Models used with mortars

  • Mortars have been used with:

– 1,2,3 phase flows in porous media – Linear elastic solid mechanics – Porescale network models

  • Prior to this research, the solution algorithm for

nonlinear problems relied on two Newton loops with a forward difference approximation.

– CG, Mixed, DG methods – Bricks, prisms, tetrahedra

  • Example:

Saturation field in two phase flow, with two subdomains.

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SLIDE 4

Center for Subsurface Modeling

Ben Ganis | GJ Mortar Algorithms | bganis@ices.utexas.edu Single Phase Mortar Theory

  • Glowinski, R., and Wheeler, M.F. 1988. Domain decomposition and mixed finite element methods

for elliptic problems. In 1st international symposium on domain decomposition methods for PDEs.

  • Arbogast, T., Cowsar, L.C., Wheeler, M.F. and Yotov, I. 2000. Mixed finite element methods on

nonmatching multiblock grids. SIAM Journal on Numerical Analysis 37 (4): 1295– 1315.

  • Arbogast, T., Pencheva, G., Wheeler, M.F., and Yotov, I. 2007. A multiscale mortar mixed finite

element method. Multiscale Modeling & Simulation 6 (1): 319–346. Forward Difference (FD) Algorithms for Nonlinear problems

  • Peszynska, M., Wheeler, M.F., and Yotov, I. 2002. Mortar upscaling for multiphase flow in porous
  • media. Computational Geosciences 6 (1): 73–100.
  • Yotov, I. 2001. A multilevel Newton–Krylov interface solver for multiphysics couplings of

flow in porous media. Numerical Linear Algebra and Applications, 8 (8): 551–570. Global Jacobian (GJ) Algorithms for Nonlinear problems

  • Ganis, B., Juntunen, M., Pencheva, G., Wheeler, M.F., and Yotov, I. 2014. A global Jacobian

method for mortar discretizations of nonlinear porous media flows. SIAM Journal on Scientific Computation 36 (2): A522–A542.

  • Ganis, B., Kumar, K., Pencheva, G., Wheeler, M.F., and Yotov, I. 2014. A global Jacobian method

for mortar discretizations of a fully-implicit two-phase flow model. Multiscale Modeling & Simulation 12 (4): 1401–1423.

  • Ganis, B., Kumar, K., Pencheva, G., Wheeler, M.F., Yotov, I. A multiscale mortar method and two-

stage preconditioner for multiphase flow using a global Jacobian approach. SPE 172990-MS.

Selected ¡References ¡on ¡Mortars ¡

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Center for Subsurface Modeling

Ben Ganis | GJ Mortar Algorithms | bganis@ices.utexas.edu

Outline

  • 1. Multiscale, Multiphase Problem Setting
  • 2. Fully-implicit two-phase model for flow in

porous media

  • 3. Global Jacobian algorithms

– Schur complements – Interface unknowns – Upwinding scheme

  • 4. Numerical results

– Strongly Heterogeneous Case – Two Rock Type Case – Non-matching Geometry Case

  • 5. Two-Stage Preconditioner and Parallel Results
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Center for Subsurface Modeling

Ben Ganis | GJ Mortar Algorithms | bganis@ices.utexas.edu

Problem Setting

  • Non-overlapping domain decomposition on spatial domain
  • Application: Multiphase flow in porous media
  • Goal: Develop simple algorithms with parallel scalability
  • Key Idea: Global linearization
  • Capillarity, gravity, and compressibility.

Interfaces Γkl Subdomains Ωk

Use mixed finite elements on structured subdomain grids Use high-order mortars (Lagrange multipliers) on non-matching interfaces

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Center for Subsurface Modeling

Ben Ganis | GJ Mortar Algorithms | bganis@ices.utexas.edu

GJ Method GJS Method FD Method New Methods Prior Method

Algorithms for nonlinear mortar problems

Time Step Nonlinear Global Newton Step Linear Global GMRES Step Time Step Nonlinear Interface FD-Newton Step Linear Interface GMRES Step Nonlinear Subdom. Newton Step Linear Subdom. GMRES Step Time Step Nonlinear Global Newton Step Linear Interface GMRES Step Linear Subdomain GMRES Step

= convergence check = forward difference approximation used

  • This algorithm uses local linearizations

for subdomain and mortar unknowns separately. – Two nested Newton-Krylov loops – Outer loop formes a numerical Jacobian with a forward difference – Requires delicate choice of four tolerances and difference parameter – Challenging to precondition outer GMRES + Allows multiple physics and multiple time steps

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SLIDE 8

Center for Subsurface Modeling

Ben Ganis | GJ Mortar Algorithms | bganis@ices.utexas.edu

GJ Method GJS Method FD Method New Methods Prior Method

Algorithms for nonlinear mortar problems

Time Step Nonlinear Global Newton Step Linear Global GMRES Step Time Step Nonlinear Interface FD-Newton Step Linear Interface GMRES Step Nonlinear Subdom. Newton Step Linear Subdom. GMRES Step Time Step Nonlinear Global Newton Step Linear Interface GMRES Step Linear Subdomain GMRES Step

= convergence check = forward difference approximation used

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Center for Subsurface Modeling

Ben Ganis | GJ Mortar Algorithms | bganis@ices.utexas.edu

Novelty of this work

Global linearization:

  • Augment linear systems to reuse codes.
  • Utilize existing preconditioners for multiscale models.
  • Simplify algorithms by having fewer nested iterations.
  • Demonstrate parallel scaling with strong nonlinearities.
  • Improve saturation with careful mobility upwinding.

Ω1 Ω2 Ω3 Γ12 Γ23 Ω1 Ω2 Ω3 Γ12 Γ23

  • =

 JΘΘ JΘΛ JΛΘ JΛΛ 

Proc1 Proc2

Ω1 Γ12 Ω2 Γ23 Ω3

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Center for Subsurface Modeling

Ben Ganis | GJ Mortar Algorithms | bganis@ices.utexas.edu

Parallel scaling, nonlinear single phase

2 4 8 16 32 64 128 1 min 5 min 20 min 1 hour 6 hours 1 day Processors (Subdomains) Wall clock time GJ GJS FD 1 2 4 8 16 32 64 128 256 512 2 min 3 min 4 min 5 min 10 min 20 min 40 min 1 hour Processors (Subdomains) Wall clock time GJ

Homogeneous, No Preconditioning Heterogeneous, AMG+ILU Preconditioner [2] B. Ganis, M. Juntunen, G. Pencheva, M.F. Wheeler, I. Yotov. A global Jacobian method for mortar discretizations of nonlinear porous media flows. SIAM Journal

  • n Scientific Computation, Vol. 36, No. 2, (2014) pp. A522-A542.

Strong scaling, O(106) elements

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Center for Subsurface Modeling

Ben Ganis | GJ Mortar Algorithms | bganis@ices.utexas.edu

Two Phase Model

∂ ∂t(φsαρα) + r · uα = qα in Ωk ⇥ (0, T] e uα = K(rpα ραg) in Ωk ⇥ (0, T] uα = krαρα µα e uα in Ωk ⇥ (0, T] e pα = pα,0 at Ω ⇥ {t = 0}, u · n = 0

  • n ∂Ω ⇥ (0, T]

pα = pΓ

α(λ1, λ2)

  • n Γ ⇥ (0, T],

uk

α · nk + ul α · nl = 0

  • n Γkl ⇥ (0, T]

sw + so = 1 pc(sw) = po pw ρα(pα) = ρref

α ecαpα.

Initial condition: Boundary condition: Lagrange multiplier: Flux continuity: Mass Balance: Saturation constraint: Capillary pressure: Slightly compressible density: Auxiliary Velocity: Darcy Law:

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Center for Subsurface Modeling

Ben Ganis | GJ Mortar Algorithms | bganis@ices.utexas.edu

Finite element discretization

Primary Unknowns: Phase Velocities: Lagrange Multipliers:

e (po, no), s (e uo, e uw, uo, uw), s (λ1, λ2)

Velocity, Pressure, Mortar Spaces:

Vh =

NΩ

M

k=1

Vk

h,

Wh =

NΩ

M

k=1

Wk

h,

M M

  • n 0 = t0 < t1 < · · · < tNT = T, with δtn = tn tn1. T

Time discretization:

mortar velocity pressure

d MH =

NΩ

M

k=1

Mkl

H.

Lowest Order Raviart- Thomas (RT0) mixed finite elements with mortars

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Center for Subsurface Modeling

Ben Ganis | GJ Mortar Algorithms | bganis@ices.utexas.edu

Fully discrete system

nα = ραsα

d mα = krαρα µα

Expanded multiscale mortar method for fully-implicit two- phase flow:

Phase concentration: Phase mobility:

Ak

α =

  • Ωk uk

α · v dx −

  • Ωk mα

uk

α · v dx = 0,

Dk

α =

  • Ωk K−1

uk

α · v dx −

  • Ωk pk

α∇ · v dx −

  • Ωk ραg · v dx +

NΩ

  • l=1,l̸=k
  • Γkl pΓ

α v · n dσ = 0,

Bk

α =

  • Ωk

φnk

α − φnn−1 α

δt w dx +

  • Ωk ∇ · uk

αw dx −

  • Ωk qαw dx = 0,

Hα =

  • Γkl
  • uk

α · nk + ul α · nl

  • µ dσ = 0.

Flux continuity equation

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Center for Subsurface Modeling

Ben Ganis | GJ Mortar Algorithms | bganis@ices.utexas.edu

Forming Residual Equations

Ak

α =

  • Ωk uk

α · v dx −

  • Ωk mα

uk

α · v dx = 0,

Dk

α =

  • Ωk K−1

uk

α · v dx −

  • Ωk pk

α∇ · v dx −

  • Ωk ραg · v dx +

NΩ

  • l=1,l̸=k
  • Γkl pΓ

α v · n dσ = 0,

Bk

α =

  • Ωk

φnk

α − φnn−1 α

δt w dx +

  • Ωk ∇ · uk

αw dx −

  • Ωk qαw dx = 0,

Hα =

  • Γkl
  • uk

α · nk + ul α · nl

  • µ dσ = 0.

Ø Express 8 unknowns as linear combinations of finite element basis functions, insert into discrete form.

e X

i=1

e pk

  • =

Nk

p

X

i=1

P k

  • ,iwk

i ,

n

X e e Uo, e Uw, Uo, Uw 2 RNu, Nu =

NΩ

X

i=1

Nk

u,

N Ø Obtain a nonlinear system for the global coefficient vectors: X , Po, No 2 RNp, d Λ1, Λ2 2 RNλ , Np =

NΩ

X

i=1

Nk

p ,

Nλ = X

1k<lNΩ

Nkl

λ

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Center for Subsurface Modeling

Ben Ganis | GJ Mortar Algorithms | bganis@ices.utexas.edu

Global nonlinear system

e Ao(e Uo, Uo, Po, No) = Aw(e Uw, Uw, Po, No) = Do(e Uo, Po, Λ1, Λ2) = Dw(e Uw, Po, No, Λ1, Λ2) = e Bo(Uo, No) = Bw(Uw, Po, No) = Ho(Uo) = Hw(Uw) =

  • Express all variables in terms of primary unknowns
  • Nonlinear system of 8 equations in 8 unknowns
  • Aux. Velocity

Darcy Velocity Mass Balance Flux Continuity

} } } }

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Center for Subsurface Modeling

Ben Ganis | GJ Mortar Algorithms | bganis@ices.utexas.edu

Forming Jacobian entries

  • Compute partial derivatives of each residual equation with

respect to each type of unknown.

  • Drop slightly compressible terms.

A1 = B @ A1

1

... ANΩ

1

1 C A , C3 = B @ C12

3

. . . C(NΩ1)NΩ

3

1 C A ,

  • Group matrices together by subdomain and interface.

… ( b Ak

3)ji ⇡ 0,

b

(Ak

1)ji =

∂Ak

  • ,j

∂ e Uo,i = (movi, vj)k , (Ak

2)ji =

∂Ak

  • ,j

∂Uo,i = (vi, vj)k , ( b Ak

3)ji =

∂Ak

  • ,j

∂Po,i = ✓✓cono µo k0

ro + coρo

µo kro ◆ wie uo, vj ◆

k

, ✓ ◆

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Center for Subsurface Modeling

Ben Ganis | GJ Mortar Algorithms | bganis@ices.utexas.edu

Global Newton step

2 6 6 6 6 6 6 6 6 6 6 4 A1 A2 A4 B1 B2 B4 C1 C2 C3 C4 D1 D2 D3 D4 D5 E1 E2 F1 F2 F3 L1 L2 3 7 7 7 7 7 7 7 7 7 7 5 2 6 6 6 6 6 6 6 6 6 6 6 4 δ e Uo δ e Uw δUo δUw δPo δNo δΛ1 δΛ2 3 7 7 7 7 7 7 7 7 7 7 7 5 = 2 6 6 6 6 6 6 6 6 6 6 4 Ao Aw Do Dw Bo Bw Ho Hw 3 7 7 7 7 7 7 7 7 7 7 5

The 8x8 fully implicit two phase global Jacobian system:

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Center for Subsurface Modeling

Ben Ganis | GJ Mortar Algorithms | bganis@ices.utexas.edu

Velocity elimination

  • We first eliminate the 4 velocities to form 1st Schur complement:

=  JΘΘ JΘΛ JΛΘ JΛΛ  δΘ δΛ

  • =

 RΘ RΛ

  • JΘΘ =

 JPoPo JPoNo JNoPo JNoNo

  • JΘΛ =

 JPoΛ1 JPoΛ2 JNoΛ1 JNoΛ2

  • JΛΘ =

 JΛ1Po JΛ1No JΛ2Po JΛ2No

  • JΛΛ =

 JΛ1Λ1 JΛ1Λ2 JΛ2Λ1 JΛ2Λ2

  • δΘ =

 δPo δNo

  • δΛ =

 δΛ1 δΛ2

  • Subdomain

unknowns Mortar unknowns

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Center for Subsurface Modeling

Ben Ganis | GJ Mortar Algorithms | bganis@ices.utexas.edu

3 Schur complements

  • 1. Can eliminate velocities to form (Θ,Λ)–Schur complement

=  JΘΘ JΘΛ JΛΘ JΛΛ  δΘ δΛ

  • =

 RΘ RΛ

  • 2. Can eliminate subdomain unknowns to form Λ–Schur complement

Here, the action of requires solving linear subdomain problems.

ΛΘJ−1 ΘΘR

  • 3. Can eliminate mortar unknowns to form Θ–Schur complement

(JΛΛ − JΛΘJ−1

ΘΘJΘΛ) δΛ = RΛ − JΛΘJ−1 ΘΘRΘ

(JΘΘ − JΘΛJ−1

ΛΛJΛΘ) δΘ = RΘ − JΘΛJ−1 ΛΛRΛ

Here, the matrix can be computed with Sparse LU or mass lumping.

ΘΛJ−1 ΛΛR

  • Starting from the saddle point system, we can form 3 different

algorithms with different character by taking Schur complements:

“GJ method” “GJS method”

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Center for Subsurface Modeling

Ben Ganis | GJ Mortar Algorithms | bganis@ices.utexas.edu

Sparsity ¡Pa4ern ¡of ¡GJ ¡Matrices ¡

Unknowns (𝜺𝑸𝑷,𝜺𝑶𝑷,𝜺𝚳𝟐,𝜺𝚳𝟑) Unknowns (𝜺𝑸𝑷,𝜺𝑶𝑷) without mass lumping Unknowns (𝜺𝑸𝑷,𝜺𝑶𝑷) with mass lumping nnz=41505 nnz=63642 nnz=44075 We will precondition this system in this work.

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Center for Subsurface Modeling

Ben Ganis | GJ Mortar Algorithms | bganis@ices.utexas.edu

Choice of interface unknowns

(Choice λ1 = pΓ

  • , λ2 = pΓ

w).

(Ckl

3 )ji =

D ηkl

j , vk i · nkE kl ,

(Ckl

4 )ji = 0,

(Dkl

4 )ji = 0,

(Dkl

5 )ji =

D ηkl

j , vk i · nkE kl .

(Ckl

3 )ji =

D ηkl

j , vk i · nkE kl ,

(Ckl

4 )ji = 0,

(Dkl

4 )ji =

D ηkl

j , vk i · nkE kl ,

(Dkl

5 )ji =

D −ηkl

j , vk i · nkE kl .

(Choice λ1 = pΓ

  • , λ2 = pΓ

c ). With this choice, pΓ w = λ1 − λ2.

  • Flexibility in choosing physical meaning of Lagrange multipliers.
  • Changes entries and condition number of GJ matrix.
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Center for Subsurface Modeling

Ben Ganis | GJ Mortar Algorithms | bganis@ices.utexas.edu

Choice of interface unknowns

(Choice λ1 = pΓ

  • , λ2 = nΓ
  • ). Using ρo, we have sw = 1 − λ2/ρo, hence

pw = λ1 − pc ✓ 1 − λ2 λ1 ◆ .

(Ckl

3 )ji =

D ηkl

j , vk i · nkE kl ,

(Ckl

4 )ji = 0,

(Dkl

4 )ji =

⌧✓ 1 − co p0

cλ2

ρo ◆ ηkl

j , vk i · nk

  • kl

, (Dkl

5 )ji =

⌧p0

c

ρo ηkl

j , vk i · nk

  • kl

.

(Dkl

4 )ji ≈

D ηkl

j , vk i · nkE kl .

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Center for Subsurface Modeling

Ben Ganis | GJ Mortar Algorithms | bganis@ices.utexas.edu

Upwinding on a single domain

pL

  • pR
  • uo

4po ⇡ pR

  • pL
  • mup
  • =

⇢ mL

  • ,

if 4po < 0 mR

  • ,

if 4po > 0 Z

Ω

mo uo · uodx ⇡

TM mup

  • ⇥

✓ hL

x

2 hy hz + hR

x

2 hy hz ◆

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Center for Subsurface Modeling

Ben Ganis | GJ Mortar Algorithms | bganis@ices.utexas.edu

Upwinding “through a mortar”

pL

  • pR
  • pλ
  • uL
  • uR
  • 4pL
  • ⇡ pλ
  • pL
  • mup,L
  • =

⇢ mL

  • ,

if 4pL

  • < 0

mλ

  • ,

if 4pL

  • > 0

Z

EL mo uL

  • · uL
  • dx ⇡

TM mup,L

  • ⇥

✓ hL

x

2 hy hz ◆ 4pR

  • ⇡ pR
  • pλ
  • mup,R
  • =

⇢ mλ

  • ,

if 4pR

  • < 0

mR

  • ,

if 4pR

  • > 0

Z

ER mo uR

  • · uR
  • dx ⇡

TM mup,R

  • ⇥

✓ hR

x

2 hy hz ◆

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Center for Subsurface Modeling

Ben Ganis | GJ Mortar Algorithms | bganis@ices.utexas.edu

What can go wrong?

  • Excessive time step cuts
  • Singular linear systems
  • Loss of nonlinear convergence
  • Loss of mass conservation

– No guarantee that pL < pλ < pR

  • r pL > pλ > pR

– May create artificial sources/sinks

  • n interfaces
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Center for Subsurface Modeling

Ben Ganis | GJ Mortar Algorithms | bganis@ices.utexas.edu

Upwinding “block-to-block”

This technique was used in enhanced velocity method and IMPES models. It is new for the fully-implicit model.

pL

  • pR
  • pλ
  • uL
  • uR
  • mup
  • =

⇢ mL

  • ,

if 4po < 0 mR

  • ,

if 4po > 0 Z

ER mo uR

  • · uR
  • dx ⇡

TM mup

  • ⇥

✓ hR

x

2 hy hz ◆ Z

EL mo uL

  • · uL
  • dx ⇡

TM mup

  • ⇥

✓ hL

x

2 hy hz ◆

Important consequences:

  • No saturation info. is

needed on interfaces.

  • No longer need Pc-1 with

extra “interface Newton”.

  • Sw is allowed to be

discontinuous even when using a continuous mortar.

4po ⇡ pR

  • pL
  • by directly projecting

ΩL|Γ ! ΩR|Γ

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Center for Subsurface Modeling

Ben Ganis | GJ Mortar Algorithms | bganis@ices.utexas.edu

Heterogeneous Case

Water Saturation Oil Velocity Magnitude

logYPERM 7 5 3 1

  • 1
  • 3
  • 5

Log Permeability

  • Challenging SPE10 industrial

benchmark case, layer 1

  • 8 subdomains, matching P0 mortars
  • Two-phase flow with gravity,

compressibility, capillary pressure

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Center for Subsurface Modeling

Ben Ganis | GJ Mortar Algorithms | bganis@ices.utexas.edu

1 2

Two Rock Type Example

Water Saturation

−

se = sw − srw 1 − srw − sro

Effective Saturation

krw = 0.9 s2

e

kro = 0.5 (1 − se)2

Relative Permeability

pc(sw) =      pd s−1/λ

c1

, if 0 ≤ se < sc1 pd s−1/λ

e

, if sc1 ≤ se ≤ sc2 pd s−1/λ

c2 1−se 1−sc2 ,

if sc2 < se ≤ 1

Capillary Pressure − − pd λ K φ rock type 1 135 psi 2.49 504 md 0.2 rock type 2 37.7 psi 3.86 52.6 md 0.2 Two Rock Types

P1 mortar H=h2/3

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Center for Subsurface Modeling

Ben Ganis | GJ Mortar Algorithms | bganis@ices.utexas.edu

Saturation Errors

pL

  • pR
  • pλ
  • uL
  • uR
  • Upwind using Lagrange multiplier
  • Max. Pointwise Error = 0.37

Upwind using adjacent subdomain values

  • Max. Pointwise Error = 0.07

Accurate integration of phase mobility can improve mass conservation and solvability

  • f linear and nonlinear systems.
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Center for Subsurface Modeling

Ben Ganis | GJ Mortar Algorithms | bganis@ices.utexas.edu

Global ¡Jacobian ¡compared ¡to ¡Forward ¡ Difference ¡Algorithm ¡

X Y Z

Y

X Y Z

XPERM 1000 100 0.1

X Y Z

logXPERM 7 6 5 4 3 2 1

  • 1
  • 2
  • 3
  • 4

FD Method Perm.

  • Intf. Newton
  • Intf. GMRES
  • Subdom. Newton

CPU Tot.

  • Avg. 1

Tot.

  • Avg. 1
  • Avg. 2

Tot.

  • Avg. 1
  • Avg. 2

Time Barrier 331 1.66 6,355 31.78 19.20 20,662 103.31 3.25 161.49 Heterog. 241 1.21 2,629 13.15 10.91 9,212 46.06 3.50 71.18 GJ Method Perm. Global Newton CPU Tot.

  • Avg. 1

Time Barrier 342 1.71 11.80 Heterog. 212 1.06 7.71

FD: best preconditioned GMRES and loose inner tolerances GJ: direct solver

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Center for Subsurface Modeling

Ben Ganis | GJ Mortar Algorithms | bganis@ices.utexas.edu

Two-­‑Stage ¡PrecondiBoning ¡

Two-Stage Preconditioners (or similar ideas) are necessary in fully-implicit multiphase models, because the linear systems have both elliptic and hyperbolic behaviors. We applied the following preconditioner to the global Jacobian multiscale mortar system:

  • Lacroix, S., Vassilevski, Y., Wheeler, M.F., 2001. Decoupling

preconditioners in the implicit parallel accurate reservoir simulator (IPARS). Numerical linear algebra with applications, 8 (8), pp. 537–549. – Four decoupling approaches are discussed:

  • Constrained Pressure Reduction (CPR)
  • Householder Reflection Decoupling ç

ç We followed this approach.

  • Quasi-IMPES Decoupling
  • True IMPES Decoupling
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Center for Subsurface Modeling

Ben Ganis | GJ Mortar Algorithms | bganis@ices.utexas.edu

More ¡Two-­‑Stage ¡References ¡

  • Vassilevski, P.S., 1984. Fast algorithm for solving a linear algebraic

problem with separable variables. Dokladi Na Bolgarskata Akademiya Na Naukite, 37 (3): 305–308.

  • Wallis, J.R., Kendall, R.P., and Little, T.E., 1985. Constrained residual

acceleration of conjugate residual methods. In SPE Reservoir Simulation Symposium, SPE 13536.

  • Cao, H., Tchelepi, H.A., Wallis, J.R., et al. 2005. Parallel scalable

unstructured CPR-type linear solver for reservoir simulation. In SPE Annual Technical Conference and Exhibition. SPE 96809.

  • Han, C. et al., 2013. Adaptation of the CPR preconditioner for efficient

solution of the adjoint equation. SPE Journal, 18(02), pp. 207–213.

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Center for Subsurface Modeling

Ben Ganis | GJ Mortar Algorithms | bganis@ices.utexas.edu

Two-­‑Stage ¡PrecondiBoning ¡for ¡GJ ¡

!!!!Θ = !!! − !!"!!!

!!!!" !!Θ = !! − !!"!!! !!!! = !!.

  • Begin with the Schur complement system for subdomain unknowns.
  • Perform Householder (QR) factorization to diagonal 2x2 blocks.

!!!!!!!!! !!Θ = !!!!!!!!! ⟺ !!!Θ = !!!!! !!!!! !!!!! !!!!! !!

!

!!! = !!! !!! = !.

  • Inside the outer gmres, get action Y = M-1 Z in a three step process:
  • 1. Solve the pressure equation YPo = gmres(HPoPo ,ZPo) with

preconditioner M1S

  • 1 to a specified tolerance.
  • 2. Update the linear residual R = Z - H[YPo,0].
  • 3. Solve the second stage equation Y = gmres(H,R) + [YPo ,0] with

preconditioner M2S

  • 1 to a specified tolerance.
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SLIDE 34

Center for Subsurface Modeling

Ben Ganis | GJ Mortar Algorithms | bganis@ices.utexas.edu

Example ¡1: ¡The ¡full ¡SPE10 ¡benchmark ¡problem ¡ with ¡mortars ¡in ¡two-­‑phase ¡model ¡

CPU cores/ Subdomains Total CPU time Total Newton Steps Taken

  • Avg. Outer

GMRES Iter. per Newton step Time Step Cuts 1×1×1=1 8331.79 51 4.88 1×1×2=2 4675.22 51 5.00 1×1×4=4 3102.14 52 5.65 1 1×2×4=8 2727.95 51 5.04 1×2×8=16 1216.14 52 5.71 1 1×4×8=32 517.69 51 5.02 1×4×16=64 618.41 109 5.71 2 1st Stage: GMRES(20), 1e–6 tolerance, 100 max iterations, M1S

  • 1 = AMG V-

cycle, 1 sweep ILU(0) smoother, coarse solve 1000x1000 with Sparse LU. 2nd Stage: GMRES(20), 1e–3 tolerance, no restarts, M2S

  • 1 = M1S
  • 1.
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Center for Subsurface Modeling

Ben Ganis | GJ Mortar Algorithms | bganis@ices.utexas.edu

Example ¡2: ¡A ¡mulBscale ¡problem ¡on ¡ ¡ non-­‑matching ¡subdomain ¡grids ¡

1st Stage: GMRES(20), 1e–3 tolerance, no restarts, M1S

  • 1 = AMG V-cycle, 1

sweep ILU(0) smoother, coarse solve 1000x1000 with Sparse LU. 2nd Stage: GMRES(1), M2S

  • 1 = 5 Gauss-Seidel iterations.
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SLIDE 36

Center for Subsurface Modeling

Ben Ganis | GJ Mortar Algorithms | bganis@ices.utexas.edu

Example ¡3: ¡A ¡heterogeneous ¡10M ¡Cell ¡Problem ¡ with ¡mortars ¡on ¡1024 ¡Processors ¡

Total ¡Bme ¡steps ¡ 1007 ¡ Total ¡Newton ¡iteraBons ¡ 1007 ¡ Total ¡outer ¡GMRES ¡iteraBons ¡ 2449 ¡ Average ¡GMRES ¡itera/ons ¡ per ¡Newton ¡step ¡ 2.43 ¡ Average ¡Newton ¡iteraBons ¡per ¡ Bme ¡step ¡ 1.00 ¡ Total ¡Bme ¡step ¡cuts ¡ ¡ Matrix ¡assembly ¡Bme ¡ 86.04 ¡ Outer ¡GMRES ¡Bme ¡ 8459.16 ¡ Householder ¡decoupling ¡Bme ¡ 42.25 ¡ Pressure ¡solve ¡GMRES ¡Bme ¡ 1394.55 ¡ Second ¡stage ¡GMRES ¡Bme ¡ 3340.99 ¡ Mass ¡lumping ¡Bme ¡ 0.05 ¡ Matrix-­‑matrix ¡mulBply ¡Bme ¡ 1206.87 ¡ Total ¡CPU ¡Bme ¡ 8571.76 ¡ 1st Stage: GMRES(20), 1e–3 tolerance, no restarts, M1S

  • 1 = AMG V-cycle, 1

sweep ILU(0) smoother, coarse solve 1000x1000 with Sparse LU. 2nd Stage: GMRES(20), 1e–3 tolerance, no restarts, M2S

  • 1 = M1S
  • 1.
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SLIDE 37

Center for Subsurface Modeling

Ben Ganis | GJ Mortar Algorithms | bganis@ices.utexas.edu

Conclusions

  • We have developed new mortar algorithms using global linearization

for single and two phase flow. – Easy to implement, fewer nested iterations and tolerances. – Inexpensive, showed parallel scalability for nonlinear problems. – Changed upwinding near interfaces for better fluid transport. – Applied two-stage preconditioner for parallel scalability.

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SLIDE 38

Center for Subsurface Modeling

Ben Ganis | GJ Mortar Algorithms | bganis@ices.utexas.edu

References

  • Ganis, B., Juntunen, M., Pencheva, G., Wheeler, M.F., and Yotov, I.
  • 2014. A global Jacobian method for mortar discretizations of

nonlinear porous media flows. SIAM Journal on Scientific Computation 36 (2): A522–A542.

  • Ganis, B., Kumar, K., Pencheva, G., Wheeler, M.F., and Yotov, I.
  • 2014. A global Jacobian method for mortar discretizations of a fully-

implicit two-phase flow model. Multiscale Modeling & Simulation 12 (4): 1401–1423.

  • Ganis, B., Kumar, K., Pencheva, G., Wheeler, M.F., Yotov, I. A

multiscale mortar method and two-stage preconditioner for multiphase flow using a global Jacobian approach. SPE 172990-MS.

Thank you!