COMPUTATIONAL ENGINEERING SCIENCES

INSTITUTE FOR

&

Adaptive Homogenization for Upscaling Heterogeneous Porous MediaGurpreet Singh†, Yerlan Amanbek†, Mary F. Wheeler††Center for Subsurface Modeling, The University of Texas at Austin

gurpreet,yerlan,mfw@ices.utexas.edu

1. Introduction

The multiscale nature of the flow and trans-port problems in porous medium is rising at-tention in pratical applications• petroleum reservoir recovery evaluations• nuclear waste disposal systems•CO2 sequestration• groundwater remediation.

the Frio Brine Pilot experiment

We propose an adaptive multiscale ap-proach to improve the efficiency andthe accuracy of numerical computationsby combining upscaling and domain de-composition methods. We use the En-hanced Velocity Mixed Finite ElementMethod (EVMFEM) as a domain decom-position approach to couple the coarseand fine subdomains [3].

2. Motivation

•Direct numerical computation is computa-tionally prohibitive, since– Heterogeneity of porous media– Complexity of dynamic systems•Capture fine scale features

– Higher resolution only near well-bore:polymer injection, gas injection

– Non-linear reactions with large variationin reaction rates

• Effective parameter computation precision

Time (days)0 50 100 150 200

Con

cent

ratio

n

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7Tracer Production Concentration

FineHomogenizedHarmonic

Tracer production history for fine and coarse (homoge-

nized and harmonic average) scale problems.

3. Model formulation

Γi,j = ∂Ωi ∩ ∂Ωj Γi = ∂Ωi \ Ω J = (0, T )

Phase Mass Conservation∇ · u = q in Ωi × J

u = −Kµ∇p in Ωi × J

Component Mass Conservation

∂(φc)

∂t+∇ · (uc−D∇c) = qαc + r(c) in Ωi × J

Boundary and Initial conditions

u · n = 0, uc · n = 0 on ∂Ω× J

c = c0, p = p0 at Ω× t = 0

Methods

The key objective of this work is to develop upscaling approaches to minimize the use offine scale information for time varying initial and boundary conditions. A fine scale flowand transport problem is solved only in specific subdomains while solving a coarse scaleproblem in the rest of the domain. This involves special treatment of sharp interfaces as-sociated with the transport equations. We define transient regions as subdomains wherespatial changes in concentration are significant. Away from the transient regions, upscal-ing is performed locally by using a numerical homogenization to obtain effective equationsat the macroscopic scale. A fine grid is then used in the transient regions and a coarsegrid everywhere else resulting in a non-matching multi-block problem.

Numerical Homogenization

Ansatz for the solution

fε(x) = f0(x, y) + εf1(x, y) + ε2f2(x, y) + ..

where fi(x, y) is a function of both x and y, periodicin y and y =

x

ε.

The gradient operator scales as ∇ = ∇x + ε−1∇y.We expect u = uε → u0 as ε→ 0.

Enhanced Velocity Mixed FEM

𝛀𝑹𝛀𝑳 𝜞

On interface Γ

• extra velocity• directly construct a flux-continuous ve-

locity approximation

4. Adaptivity criteria

•Criteria 1 (gradient in time):

Ωf =

x : |cn(x)− cn−1(x)| > εtadap

•Criteria 2 (gradient in space): We define

Ωneighbor(x) = y : y ∈ Ej, |∂Ei ∩ ∂Ej| 6= ∅, if x ∈ Ei.

The criteria can then be defined as,

Ωf =

x : max|cn(x)− cn(y)| > εsadap ∀y ∈ Ωneighbor(x)

5. Numerical Experiments

5.1 Homogeneous permeability

0 10 20 30 40 50 60 70 80 90 100 1100

5

10

15

20

25

30Concentration at time 100 days

0

0.2

0.4

0.6

0.8

1

0 10 20 30 40 50 60 70 80 90 100 1100

5

10

15

20

25

30Concentration at time 100 days

0

0.2

0.4

0.6

0.8

1

Figure 1: Concentration profiles at 100days for fine (top) and adaptive (bottom)approaches

Time (days)0 50 100 150 200

Con

cent

ratio

n

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7Tracer Production Concentration

Adaptive 2HomogenizedFineAdaptive 1

Figure 2: Tracer concentration history atproduction well for a homogeneous per-meability distribution of 50 mD

5.2 Benchmark datasets: SPE 10

0 10 20 30 40 50 60 70 80 90 100 1100

5

10

15

20

25

30Concentration at time 100 days

0

0.2

0.4

0.6

0.8

1

0 10 20 30 40 50 60 70 80 90 100 1100

5

10

15

20

25

30Concentration at time 100 days

0

0.2

0.4

0.6

0.8

1

0 10 20 30 40 50 60 70 80 90 100 1100

5

10

15

20

25

30Concentration at time 100 days

0

0.2

0.4

0.6

0.8

1

Figure 3: Concentration profiles at 100 daysfor coarse (top) , fine (middle) and adaptive(bottom) approaches

Fine Permeability Distribution

y

x

Figure 4: Fine and Coarse scale perme-ability distributions for SPE 10 layer 37

Time (days)0 50 100 150 200

Con

cent

ratio

n

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4Tracer Production Concentration

FineAdaptive 1HomogenizedAdaptive 2

Figure 5: Tracer concentration history at the production well for SPE layer 37.

6. Conclusions

We developed an adaptive multiscale scheme using local numerical homogenization and EV MFEM forupscaling single phase flow and transport in a heterogeneous porous media. The numerical results ondifferent layers of SPE10 also indicate that an upscaling based solely upon numerical homogenization is ingood agreement with the fine scale solution for a Gaussian or periodic permeability distribution. However,for highly channelized for layered permeability distributions the results deviate substantially. The adaptivemultiscale scheme, on the other hand, is in good agreement for Gaussian, periodic, and layered perme-ability distributions and is approximately faster than the fine scale simulation for a tracer slug injection of 50days in all the above numerical tests. It is important to note that a tracer slug injection is specifically chosento test adaptivity since there are two transient regions at the front and back of the injected slug. The accu-racy of the adaptive multiscale approach presented here, compared to the fine scale solution, is dependenton the nature of physical process and validity of the adaptivity criteria in capturing fine scale physics. Anoptimal tolerance for the adaptivity criteria is chosen to reduce computational cost without substantial lossin solution accuracy.

References

[1] Y. Amanbek, G. Singh, M.F. Wheeler, and H. van Duijn. Adaptive numerical homogenization for upscaling single phase flowand transport. Technical report. ICES Report 17-12, June, 2017.

[2] G. Singh, Y. Amanbek, and M. F. Wheeler. Adaptive numerical homogenization for non-linear multiphase flow and transport.Technical report. ICES Report 17-13, June, 2017.

[3] John A Wheeler, Mary F Wheeler, and Ivan Yotov. Enhanced velocity mixed finite element methods for flow in multiblockdomains. Computational Geosciences, 6(3-4):315–332, 2002.