shuyu sun earth science and engineering program kaust presented at the 2009 annual utam meeting,...

Shuyu SunEarth Science and Engineering program

KAUST

Presented at the 2009 annual UTAM meeting, 2:05-2:40pm January 7, 2010 at the Sutton Building, University of Utah, Salt Lake City, Utah

Energy and Environment Problems

Single Phase Flow in Porous Media

• Continuity equation – from mass conservation

• Thermodynamic model

• For impressible fluid (constant density):

• Still need one more equation

€

∂ρ∂t

+∇ ⋅ ρu( ) = qm , (x, t)∈ Ω × (0,T]

€

ρ =ρ(T,P)

€

∇⋅u = q (x, t)∈ Ω × (0,T]

Relate velocity with pressure: Darcy's law

• Experiment by Henry Darcy (1855–1856)

Darcy's law

• Can be derived from the Navier-Stokes equations via homogenization.

• It is analogous to – Fourier's law in the field of heat conduction,– Ohm's law in the field of electrical networks,– Fick's law in diffusion theory.

• In 3D:

Incompressible Single Phase Flow

• Continuity equation

• Darcy’s law

• Boundary conditions:

],0(),( Ttxq u

],0(),( Ttxp K

u

€

p = pB (x, t)∈ ΓD × (0,T]

u ⋅n = uB (x, t)∈ ΓN × (0,T]

Transport in Porous Media

• Transport equation

• Boundary conditions

• Initial condition

• Dispersion/diffusion tensor

],0(),()()( * Ttxcrqccct

c

uDu

€

uc − D∇c( ) ⋅n = cBu ⋅n t ∈ (0,T], x ∈ Γin (t)

−D∇c( ) ⋅n = 0 t ∈ (0,T], x ∈ Γout (t)

xxcxc )()0,( 0

)()()( uEIuEuIuD tlmD

Numerical Methods for Flow & Transport

• Challenge #1: Require the numerical method to be: – Locally conservative for the volume/mass of fluid (flow

equation) – Locally conservative for the mass of species (transport

equation) – Provides fluxes that is continuous in the normal direction

across the entire domain.

• Methods that are not locally conservative without post-processing– Point-Centered Finite Difference Methods– Continuous Galerkin Finite Element Methods – Collocation methods– ……

Example: importance of local conservation


• Challenge #2: Fractured Porous Media – Different spatial scale: fracture much smaller– Different temporal scale: flow in fracture much faster

• Solutions:– Mesh adaptation for spatial scale difference– Time step adaptation for temporal scale difference

Example: flow/transport in fractured media


Locally refined mesh:

FEM and FVM are better than FDfor adaptive meshes and complex geometry


CFL condition requires much smaller time step in fractures than in matrix: adaptive time stepping.


• Challenge #3: Sharp fronts or shocks – Require a numerical method with little numerical diffusion – Especially important for nonlinearly coupled system, with

sharp gradients or shocks easily being formed

• Solutions:– Characteristic finite element methods – Discontinuous Galerkin methods

Example: Comparison of DG and FVM

Advection of an injected species from the left boundary under constant Darcy velocity. Plots show concentration profile at 0.5 PVI.

Upwind-FVM on 40 elements Linear DG on 40 elements


Flow in a medium with high permeability region (red) and low permeability region (blue) with flow rate specified on left boundary. Contaminated fluid flood into clean media.


Advection of an injected species from the left. Plots show concentration profiles at 3 years (0.6 PVI).

FVM Linear DG

Numerical Method for Flow & Transport

• Challenge #4: Time dependent local phenomena– For example: moving contaminant plume

• Solutions:– Dynamic mesh adaptation

• Based on conforming mesh adaptation• Based on non-conforming mesh adaptation

Adaptive DG methods – an example

• Sorption occurs only in the lower half sub-domain,

• SIPG is used.

Adaptive DG example (cont.)

Ani

sotr

opic

mes

h ad

apta

tion


Est

imat

ors

usin

g hi

erar

chic

bas

es


L2(

L2)

Err

or E

stim

ator

s

A Posteriori Error Estimators

• Residual based – L2(L2)– L2(H1)

• Implicit – Solve a dual problem, can give estimates on a target

functional– Disadvantages: computational costly and not flexible– Advantages: More accurate estimates

• Hierarchical bases – Brute-force: difference between solutions of two

discretizations (most expensive)– Local problems-based – Advantage: can guide anisotropic hp-adaptivity

• Superconvergence points-based – Difficult for unstructured and non-conforming meshes

A posteriori error estimates

• Residuals– Interior residuals

– (Element-)boundary residuals

DGDGDG

DGDGI CC

t

CCMrqCR

Du)(*

outhDG

inhDGDG

B

hDG

B

DirihBDG

hDG

B

xC

xCCc

xC

R

xcC

xCR

,

,1

,

0

nD

nDuu

nD

A posteriori error estimate in L2(L2) for SIPG

2/1

2

)( 22

hEELL

DG KcCΕ

ELLB

ELLB

ELLBELLI

EE

Rr

hR

r

h

Rrhr

hR

r

h

2

))((13

32

))((13

3

2

))((0

2

))((4

42

2222

2222

2

1

2

1

Proof Sketch: Compare with L2 projection; Cauchy-Schwarz; Properties of cut-off operator; Approximation results; Inverse and Gronwell’s inequalities; Relation of residue and error

Dynamic mesh adaptation with DG

• Nonconforming meshes– Effective implementation of mesh

adaptation,– Elements will not degenerate unless using

anisotropic refinement on purpose.

• Dynamic mesh adaptation – Time slices = a number of time steps; only

change mesh for time slices. – Refinement + coarsening number of

elements remain constant.

Concentration projections during dynamic mesh modification

• Standard L2 projection used– Computation involved only in elements being

coarsened

• L2 projection is a local computation for discontinuous spaces– This results in computational efficiency for DG– L2 projection is a global computation for CG

• L2 projection is locally mass conservative– This maintains solution accuracy for DG– Interpolation or interpolation-based projection

used in CG is NOT locally conservative

Adaptive DG example (quads)L

2(L

2) E

rror

Est

imat

ors

for

SIP

G

Adaptive DG (with triangles)

L2(

L2)

Err

or E

stim

ator

s on

Tri

angl

es

Initial mesh


L2(

L2)

Err

or E

stim

ator

s on

Tri

angl

es

T=0.5

T=1.0


L2(

L2)

Err

or E

stim

ator

s on

Tri

angl

es

T=1.5

T=2.0

Adaptive DG example in 3DL

2(L

2) E

rror

Est

imat

ors

on 3

D

T=1.5

T=2.0

T=0.1 T=0.5 T=1.0

ANDRA-Couplex1 case

Background– ANDRA: the French National Radioactive Waste Management Agency

– Couplex1 Test Case• Nuclear waste management: Simplified 2D Far Field model• Flow, Advection, Diffusion-dispersion, Adsorption

Challenges– Parameters are highly varying

• permeability; retardation factor; effective porosity; effective diffusivity

– Very concentrated nature of source• concentrated in space • concentrated in time

– Long time simulation• 10 million years

– Multiple space scales• Around source / Far from source

– Multiple time scales• Short time behavior (Diffusion dominated) • Long time behavior (Advection dominated)

ANDRA-Couplex1 case (cont.)

200k years

2m years

Compositional Three-Phase Flow

• Mass Conservation (without molecular diffusion)

• Darcy’s Law

gowPkr ,,, ρ

gKu

gow

ii xc,,

,

uU

Numerical Modeling for Flow & Transport

• Challenge #5: Importance of capillarity – Capillary pressure usually ignored in compositional flow

modeling– Even the immiscible two-phase flow or the black oil model

usually assumes only a single capillary function (i.e. assuming a single uniform rock)

• Two-dimensional 400x200m^2 domain • Contains a less-permeable (K=1md) rock in the center

of the domain while the rest has K=100md. Isotropic permeability tensor used.

• Porosity = 0.2 • Densities: 1000 kg/m^3 (W) and 660 kg/m^3 (O)• Viscosities: 1 cp (W) and 0.45 cp (O) • Inject on the left edge, and produce on the right edge• Injection rate: 0.1 PV/year• Initial water saturation: 0.0; Injected saturation: 1.0

Example: Reservoir Description

• Relative permeabilities (assuming zero residual saturations):

• Capillary pressure

Reservoir Description (cont.)

2,,1, mSSSkSk wwem

wernmwerw

bars50 and5,,log)( cwwewecwec BSSSBSp

K=100md

K=1md

Discretization • DG-MFE-Iterative • Pressure time step: 10years / 1000 timeSteps• Saturation time step = 1/100 pressure time step• Mesh: 32x64 uniform rectangular grid:

Comparison: if ignore capillary pressure …

Saturation at 10 years: Iter-DG-MFE

With nonzero capPres

With zero capPres


• Challenge #6: Discontinuous saturation distribution– Saturation usually is discontinuous across different rock

type, which is ignored in many works in literature – When permeability changes, the capillary function usually

also changes!

• Solutions: – Discontinuous Galerkin methods

Saturation at 3 years

Iter-DG-MFE Simulation

Notice that Sw is continuous within each rock, but Sw is discontinuous across the two rocks

Water pressure at 10 years

Iter-DG-MFE Simulation (pressure unit: Pa)

Notice that Pw is continuous within the entire domain.

Capillary pressure at 10 years

Iter-DG-MFE Simulation (pressure unit: Pa)

Notice that Pc is continuous within the entire domain.


• Challenge #7: Multiscale heterogeneous permeability

– Fine scale permeability has pronounced influence on coarse scale flow behaviors

– Direct simulation on fine scale is intractable with available computational power

• Solutions: – Upscaling schemes – Multiscale finite element methods

Recall: DG scheme for flow equation

• Bilinear form

• Linear functional

• Scheme: seek such that

hDD

hhh

Eee

e

e

ee e

ee e

Eee e

Eee e

TEE

wph

pK

spK

pK

spK

pK

pa

]][[

]}[{]}[{),(

form

form

nn

nn

ND e

e Be

e Be upK

sql

nform),()(

)( hkh TDp

)()(),( hkh TDvvlvpa

IIPG SIPG,0

NIPG0

DG-OBB0

IIPG0

NIPG DG,-OBB1

SIPG1

form

s

DG on two meshes

• Fine mesh

• Coarse mesh

( ): ( , ) ( ) ( )h r h h r hp D T a p v l v v D T

( ): ( , ) ( ) ( )H R H H R Hp D T a p v l v v D T

Space decomposition

• Introduce

• Solution

( ) ( )r h R H fD T D T V

( ), :

( , ) ( ) ( , ) ( )

( , ) ( ) ( , )

H R H f f

H f R H

f H f

p D T p V

a p v l v a p v v D T

a p v l v a p v v V

fV

.h H fp p p

Closure Assumption

• Introduce

• Two-scale solution

0 : 0,f f HEV v V v E T

0 0

0

0 0

( ), :

( , ) ( ) ( , ) ( )

( , ) ( ) ( , )

H R H f f

H f R H

f H f

p D T p V

a p v l v a p v v D T

a p v l v a p v v V

0fV

0.MS H fp p p

Implementation

• Multiscale basis functions:For each

• Multiscale approximation space:

• Two-scale DG solution

0( ) : ( )MS f H H R HV v v v D T

: ( , ) ( ) ,MS MS MS MSp V a p v l v v V

0 0

0 0

( ) :

( ( ), ) ( ) ( , )

f H f

f H H f

v v V

a v v v l v a v v v V

( )H R Hv D T

Other Closure Options

• Local problems for solving multiscale basis functions need a closure assumption.

• In previous derivation, we strongly impose zero Dirichlet boundary condition on local problems.

• Other options: – Weakly impose zero Dirichlet boundary condition on local

problems. – Strongly impose zero Neumann boundary condition on local

problems.– Weakly impose zero Neumann boundary condition on local

problems.– Combination of zero Neumann and zero Dirichlet.

Comparison with direct DG

• Memory requirement– Direct DG solution in fine mesh: – Multiscale DG solution

• Computational time – Direct DG solution in fine mesh:

– Multiscale DG solution

( )d

d

rO

h

( ) ( )( / )

d d

d d

R rO O

H h H

1 system of ( ) dofsd

d

rO

h

( ) systems of ( )dofs + 1 system of ( )dofs( / )

d d d

d d d

H r RO O O

h h H H

Example

• Conductivity:

• Boundary conditions: – Left: p=1; Right: p=0; top & bottom: u=0.

• Discretization: – R=r=1; – Coarse mesh 16x16; Fine mesh 256x256

Example (cont.)

Coarse DG solution Brute-force Fine DG solution

Example (cont.)

Multiscale DG solution Brute-force Fine DG solution

Future work

• Multiscale DG methods for compositional multiple-phase flow in heterogeneous media,

• Stochastic PDE simulations,• Multigrid solver for DG (including p-multigrid), • Other future works:

– Automatically adaptive time stepping,– Implicit a posteriori error estimators, – Fully automatically hp-adaptivity for DG, – A posteriori estimators for coupled reactive transport

and flow.

shuyu sun earth science and engineering program kaust presented at the 2009 annual utam meeting,...

Documents

flow transportchallenge

flow rate

adaptive time

dynamic mesh adaptationbased

fractured mediaexample

left boundary

injected species

little numerical diffusion