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Transcript of 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
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]
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– ……
Numerical Methods for Flow & Transport
• 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
Example: flow/transport in fractured media
CFL condition requires much smaller time step in fractures than in matrix: adaptive time stepping.
Numerical Methods for Flow & Transport
• 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
Example: Comparison of DG and FVM
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.
Example: Comparison of DG and FVM
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.
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
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)
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
Numerical Modeling for Flow & Transport
• 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
Saturation at 5 years
Iter-DG-MFE Simulation
Notice that Sw is continuous within each rock, but Sw is discontinuous across the two rocks
Saturation at 10 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.
Numerical Modeling for Flow & Transport
• 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
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.