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Mar 2, 2017 - Multiscale methods for flow and transport in porous media. Yerlan Amanbek,. Gurpreet Singh, Gergina Pencheva, Tim Wildey, Mary Wheeler.

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: [email protected]

Multiscale methods for flow and transport in porous media

March 2, 2017

1 / 31

Outline Motivation Numerical Homogenization Enhanced Velocity Mixed FEM a priori error estimates for Parabolic pde Adaptive method Summary and Future work

Yerlan Amanbek: [email protected]

Multiscale methods for flow and transport in porous media

March 2, 2017

2 / 31

Motivation

Enhanced Oil Recovery

CO2 sequestration: Frio field

Groundwater remediation: Hanford site

Environmental contamination: Oak Ridge

Yerlan Amanbek: [email protected]

Multiscale methods for flow and transport in porous media

March 2, 2017

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Introduction: Frio Field

Yerlan Amanbek: [email protected]

Multiscale methods for flow and transport in porous media

March 2, 2017

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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.

Yerlan Amanbek: [email protected]

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: [email protected]

Multiscale methods for flow and transport in porous media

March 2, 2017

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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: [email protected]

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: [email protected]

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

Yerlan Amanbek: [email protected]



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: [email protected]

Multiscale methods for flow and transport in porous media

March 2, 2017

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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].

Yerlan Amanbek: [email protected]

Multiscale methods for flow and transport in porous media

March 2, 2017

4 / 31

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: [email protected]

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: [email protected]

Multiscale methods for flow and transport in porous media

March 2, 2017

6 / 31

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