Multiscale methods for flow and transport in porous media
Yerlan Amanbek, Gurpreet Singh, Gergina Pencheva, Tim Wildey, Mary Wheeler SIAM CSE 17 Center for Subsurface Modeling
Yerlan Amanbek:
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Multiscale methods for flow and transport in porous media
March 2, 2017
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Outline Motivation Numerical Homogenization Enhanced Velocity Mixed FEM a priori error estimates for Parabolic pde Adaptive method Summary and Future work
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Multiscale methods for flow and transport in porous media
March 2, 2017
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Motivation
Enhanced Oil Recovery
CO2 sequestration: Frio field
Groundwater remediation: Hanford site
Environmental contamination: Oak Ridge
Yerlan Amanbek:
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Multiscale methods for flow and transport in porous media
March 2, 2017
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Introduction: Frio Field
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Multiscale methods for flow and transport in porous media
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Characterizing Reservoir Heterogeneity Sandstone reservoirs Periodic deposition due to flooding of river beds Shale layer marks the end of one deposition cycle
Idealize as a periodic porous medium Identify meso-scale periodicity from well log data Characterize period High permeability Low permeability
Solve local period problem to estimate up-scaled field scale permeability We intend to use multiscale approach.
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Figure: Frio pilot injection well Permeability vs Depth
Multiscale methods for flow and transport in porous media
March 2, 2017
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Fine scale problem Let Ω ∈ Rn be domain
Solution of elliptic equation with rapidly oscillatory coefficient
0.25
( −∇ · (K ε ∇u ε ) = f for x ∈ Ω uε = 0 for x ∈ ∂Ω
0.2
Solution
0.15
Assumptions: ∃α, β : α|ξ|2 ≤ Y = (0, 1)
PN
i ,j=1
Fine scale, u Coarse scale, u
0.1
f ∈ L 2 (Ω), Ω ∈ Rd bounded, ∞ K ε ∈ Lper (Y , Rdxd ) (sufficiently regular) Kiε,j (y)ξi ξj ≤ β|ξ|2 , ∀ξ ∈ Rd .
0.05
d
0 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
x
Yerlan Amanbek:
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Multiscale methods for flow and transport in porous media
March 2, 2017
6 / 31
x
Two-Scale asymptotic expansions Ansatz for the solution
y
ε = 0.5
2
x
uε (x ) = u0 (x , y) + εu1 (x , y) + ε u2 (x , y) + .. Here, ui (x , y) is a function of both x and y, periodic in y x and y = . The gradient operator scales as
y
ε = 0.2
∇ = ∇ x + ε− 1 ∇ y
x
ε
y
ε = 0.05
x
We expect u = uε → u0
as
ε → 0.
y
ε→0
Yerlan Amanbek:
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Multiscale methods for flow and transport in porous media
March 2, 2017
7 / 31
Theorem 1 ( −∇ · K ( xε )∇u ε (x ) = f uε = 0 for x ∈ ∂Ω uε * u
in
H01 (Ω)
x∈Ω
for
uε → u i ,j
Keff =
R
j=1,2,..d .
L 2 (Ω)
∂ χj Ki ,j (y)+ (y)dy K ( y ) i , n Y n =1 ∂ yn
( −∇ · (Keff ∇u ) = f in Ω u=0 on ∂Ω
in
d P
GrigoriosA.Pavliotis,A.M .Stuart ,2008
1 χj ∈ Cper (Y ),
Z
χj dy = 0 with
Y ∂ Ki ,j −∇ · (K (y)∇χj ) = ∂ yi Yerlan Amanbek:
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Multiscale methods for flow and transport in porous media
in
Y March 2, 2017
8 / 31
Arithmetic & Harmonic Average "
10 0
Kr =
¯11 = k
−1
hk11 i k21 k11
" & Kl =
5
h
1
h ¯12 = k
0
#
¯12 = k
k12
100
0
0
50
#
i
k11 −1
hk11 i
Figure: Unit Cell Permeability Distribution i
−1 hk11 i
k22 = h
k21 k11
ih
k12
i
1
−1 k11 hk11 i
+ hk22 −
k21 k12 k11
i
Computation result:
"
1
−1 Keff = hk11 i 0
= Harm avg 0 hk22 i 0
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0 Arith avg
Keff =
#
18.1818
0
0
27.5
Multiscale methods for flow and transport in porous media
March 2, 2017
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Homogenization: An Upscaling Approach
Figure: An upscaling approach
Assume Darcy’s law is valid for unit cell(mesoscale) Characteristic length scales:Lunit /Lreservoir = ε > ε.
Yerlan Amanbek:
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Multiscale methods for flow and transport in porous media
March 2, 2017
3 / 31
Navier-Stokes equations vs Darcy’s law
? (a) C. Navier (1785-1836) & G. Stokes (1819-1903)
(b) Henry Darcy (1803-1853)
Homogenization
Navier-Stokes equations =================⇒ Darcy’s law equations Source [1].
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Multiscale methods for flow and transport in porous media
March 2, 2017
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Model formulation We consider a single-phase flow model for pressure p and the velocity u:
∂ (φci ρ) + ∇ · (ρuci − ρDi ∇ci ) = qi + φri i = 1, ..Nc , ∂t ∂ (φρ) + ∇ · ρ u = q ∂t u = −K (∇p − ρg∇H ) in Ω, p = pb on ∂Ω × J p(x , t ) = p0 (x )
on Ω × {0},
ci0 (x )
on Ω × {0},
ci (x , t ) =
where Ω ∈ Rd (d = 1, 2 or 3) is domain and K is a symmetric, uniformly positive definite tensor representing the permeability divided by the viscosity.
Yerlan Amanbek:
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Multiscale methods for flow and transport in porous media
March 2, 2017
5 / 31
Theorem 2 Consider u ε (x ) and u (x ) as in Theorem 1. Assume that f ∈ L 2 (Ω) and ∂Ω is sufficiently smooth so that u ∈ H 2 (Ω) ∩ H01 (Ω), and that the coefficient matrix K 1 is such the cell problem has solution χ ∈ Cper (Y ). Then,
lim ||u ε (x ) − u (x ) − εχ
ε→0
x ε
· ∇u (x ) ||H 1 (Ω) = 0.
Remark. ∞ 1 For χ ∈ Cper (Y ), K (y) has to be sufficiently regular; K ∈ Cper (Y , Rdxd ) is more than sufficient.
Yerlan Amanbek:
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Multiscale methods for flow and transport in porous media
March 2, 2017
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