Adaptive Homogenization for Upscaling Heterogeneous Porous ... · UT Adaptive Homogenization for...
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COMPUTATIONAL ENGINEERING SCIENCES INSTITUTE FOR & Adaptive Homogenization for Upscaling Heterogeneous Porous Media Gurpreet 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 and the accuracy of numerical computations by combining upscaling and domain de- composition methods. We use the En- hanced Velocity Mixed Finite Element Method (EVMFEM) as a domain decom- position approach to couple the coarse and 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 variation in reaction rates • Effective parameter computation precision Time (days) 0 50 100 150 200 Concentration 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Tracer Production Concentration Fine Homogenized Harmonic Tracer production history for fine and coarse (homoge- nized and harmonic average) scale problems. 3. Model formulation Γ i,j = ∂ Ω i ∩ ∂ Ω j Γ i = ∂ Ω i \ Ω J = (0,T ) Phase Mass Conservation ∇· u = q in Ω i × J u = - K μ ∇p in Ω i × J Component Mass Conservation ∂ (φc) ∂t + ∇· (uc - D∇c)= q α ˆ c + r (c) in Ω i × J Boundary and Initial conditions u · n =0, uc · n =0 on ∂ Ω × J c = c 0 , p = p 0 at Ω ×{t=0} Methods The key objective of this work is to develop upscaling approaches to minimize the use of fine scale information for time varying initial and boundary conditions. A fine scale flow and transport problem is solved only in specific subdomains while solving a coarse scale problem 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 where spatial changes in concentration are significant. Away from the transient regions, upscal- ing is performed locally by using a numerical homogenization to obtain effective equations at the macroscopic scale. A fine grid is then used in the transient regions and a coarse grid everywhere else resulting in a non-matching multi-block problem. Numerical Homogenization Ansatz for the solution f ε (x)= f 0 (x, y )+ εf 1 (x, y )+ ε 2 f 2 (x, y )+ .. where f i (x, y ) is a function of both x and y , periodic in y and y = x ε . The gradient operator scales as ∇ = ∇ x + ε -1 ∇ y . We expect u = u ε → u 0 as ε → 0. Enhanced Velocity Mixed FEM On interface Γ • extra velocity • directly construct a flux-continuous ve- locity approximation 4. Adaptivity criteria • Criteria 1 (gradient in time): Ω f = n x : |c n (x) - c n-1 (x)| > tadap o • Criteria 2 (gradient in space): We define Ω neighbor (x)= {y : y ∈ E j , |∂E i ∩ ∂E j |6 = ∅, if x ∈ E i .} The criteria can then be defined as, Ω f = x : max|c n (x) - c n (y )| > sadap ∀y ∈ Ω neighbor (x) 5. Numerical Experiments 5.1 Homogeneous permeability 0 10 20 30 40 50 60 70 80 90 100 110 0 5 10 15 20 25 30 Concentration at time 100 days 0 0.2 0.4 0.6 0.8 1 0 10 20 30 40 50 60 70 80 90 100 110 0 5 10 15 20 25 30 Concentration at time 100 days 0 0.2 0.4 0.6 0.8 1 Figure 1: Concentration profiles at 100 days for fine (top) and adaptive (bottom) approaches Time (days) 0 50 100 150 200 Concentration 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Tracer Production Concentration Adaptive 2 Homogenized Fine Adaptive 1 Figure 2: Tracer concentration history at production 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 90 100 110 0 5 10 15 20 25 30 Concentration at time 100 days 0 0.2 0.4 0.6 0.8 1 0 10 20 30 40 50 60 70 80 90 100 110 0 5 10 15 20 25 30 Concentration at time 100 days 0 0.2 0.4 0.6 0.8 1 0 10 20 30 40 50 60 70 80 90 100 110 0 5 10 15 20 25 30 Concentration at time 100 days 0 0.2 0.4 0.6 0.8 1 Figure 3: Concentration profiles at 100 days for 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 Concentration 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 Tracer Production Concentration Fine Adaptive 1 Homogenized Adaptive 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 for upscaling single phase flow and transport in a heterogeneous porous media. The numerical results on different layers of SPE10 also indicate that an upscaling based solely upon numerical homogenization is in good 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 adaptive multiscale 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 50 days in all the above numerical tests. It is important to note that a tracer slug injection is specifically chosen to 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 dependent on the nature of physical process and validity of the adaptivity criteria in capturing fine scale physics. An optimal tolerance for the adaptivity criteria is chosen to reduce computational cost without substantial loss in solution accuracy. References [1]Y. Amanbek, G. Singh, M.F. Wheeler, and H. van Duijn. Adaptive numerical homogenization for upscaling single phase flow and 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 multiblock domains. Computational Geosciences, 6(3-4):315–332, 2002.
Transcript of Adaptive Homogenization for Upscaling Heterogeneous Porous ... · UT Adaptive Homogenization for...
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,[email protected]
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.
