Matching for Noisy Data of Multiple Types Gradient-based...
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Gradient-based Pareto Optimal History Matching for Noisy Data of Multiple Types Oleg Volkov (Stanford University), Vladislav Bukshtynov (Florida Tech), Louis J. Durlofsky, Khalid Aziz (Stanford University) A BSTRACT The advantages of the simultaneous integra- tion of production and time-lapse seismic data for history matching (HM) have been demon- strated in multiple studies. Production data provide accurate observations at specific spa- tial locations (wells), while seismic data en- able global, though filtered/noisy, estimates of state variables. In this work we present an ef- ficient computational tool for bi-objective HM, in which data misfits for both production and seismic measurements are minimized using an adjoint-gradient approach. This enables us to obtain a set of Pareto optimal solutions defining the optimal trade-off between production and seismic data conflicting misfits. The impact of noise structure and noise level on Pareto opti- mal solutions is investigated in detail. We dis- cuss the existence of the “best” trade-off solu- tion, or least-conflicting posterior model, which corresponds to the history matched model that is expected to provide least-conflicting forecast of future reservoir performance. The overall framework is successfully applied in 2D and 3D compositional simulation problems. P ARETO O PTIMAL HM Reservoir PDE (states x, model parameters u): g(x, u)= 0, x(t 0 )= x 0 . Multiobjective function to be minimized: J mo = α J * prod · J prod J 0 prod + 1 - α J * seism · J seism J 0 seism ,α ∈ [0, 1], production data (well rates ˜ q j,p ) J prod (x, u)= N obs X k=1 N q well X j =1 N p X p=1 C j,p (q j,p - ˜ q j,p ) 2 , time-lapse seismic data (phase saturation ˜ s j,p ) J seism (x, u)= N obs X k=1 N block X j =1 N p X p=1 C p (s j,p - ˜ s j,p ) 2 . C ASE S TUDY #1: 2D M ODEL 1 2 3 4 5 6 7 8 9 10 11 12 13 10 20 30 40 5 10 15 20 25 30 35 40 45 1 2 3 4 5 6 7 8 9 Geometry: uniform grid, 45 × 45 blocks Geology: in SGeMS, spherical variogram & target histogram Uncertain parame- ters: permeability Parameterization: PCA on 1000 realizations Model: compositional (6 components), 4 injec- tors + 9 producers (BHP controlled) History: 200 days, rates & seismic with noise Fluid components: same for 2D/3D models Component CO 2 C 1 C 2 C 3 C 4 C 10 Initial (%) 1 20 20 15 10 34 Injection (%) 100 – – – – – A CKNOWLEDGMENTS Stanford University Reservoir Simulation Research (SUPRI-B), Smart Fields Consortia, Tapan Mukerji (ERE, Stanford) B I - OBJECTIVE HM R ESULTS 10 -3 10 -2 10 -1 10 -2 10 -1 production part seismic part 2% 5% 0% 15% 30% 50% noise: 5% prod + 0% seism noise: 5% prod + 30% seism noise: 2% prod + 0% seism noise: 2% prod + 15% seism noise: 2% prod + 30% seism noise: 2% prod + 50% seism Noise in production and seismic data by stan- dard deviations of measurement error: σ q = ( σ q,min , γ q · q p <σ q,min , γ q · q p , γ q · q p ≥ σ q,min , σ s = γ s (1 - φ)(1 - s true g ). Fig. 1: Pareto optimal solutions for different levels of noise in production and seismic data (2D model). Case study #1 Predictions using HM models: • α =0: only seismic data, • α =0.5, • α =1: only production data, • α =0.826: best trade-off so- lution determined at a point with the highest curvature Fig. 2: (a-f) Predictions for CO 2 injection (a, b) and production (c–f) rates with 2% noise in production and 50% in seismic data. (g-i) Error in predicted sat- uration ks g - s true g k L 2 ks true g k L 2 with (g) no noise, (h) 5% noise in production data and (i) 2% noise in production and 50% in seismic data. (a) well #2 (injector) 0 100 200 300 400 500 600 0 200 400 600 800 1000 days CO 2 injection rate (d) well #7 (producer) 0 100 200 300 400 500 600 0 100 200 300 400 500 days CO 2 production rate (g) no noise 0 100 200 300 400 500 600 0 0.1 0.2 0.3 0.4 0.5 days saturation fit initial α = 0.0 (seism) α = 0.5 α = 1.0 (prod) (b) well #3 (injector) 0 100 200 300 400 500 600 0 200 400 600 800 1000 1200 days CO 2 injection rate (e) well #9 (producer) 0 100 200 300 400 500 600 0 100 200 300 400 500 600 700 days CO 2 production rate (h) noise 5% production & 0% seismic 0 100 200 300 400 500 600 0 0.1 0.2 0.3 0.4 0.5 days saturation fit initial α = 0.0 (seism) α = 0.5 α = 1.0 (prod) best trade-off (c) well #5 (producer) 0 100 200 300 400 500 600 0 100 200 300 400 500 600 700 days CO 2 production rate (f) well #10 (producer) 0 100 200 300 400 500 600 0 500 1000 1500 2000 2500 days CO 2 production rate true initial α = 0.0 (seism) α = 0.5 α = 1.0 (prod) best trade-off (i) noise 2% production & 50% seismic 0 100 200 300 400 500 600 0 0.1 0.2 0.3 0.4 0.5 days saturation fit initial α = 0.0 (seism) α = 0.5 α = 1.0 (prod) best trade-off 10 -1 10 0 10 -0.4 10 -0.2 10 0 10 0.2 production part seismic part 10 -1 10 0 10 -0.3 10 -0.2 10 -0.1 production part seismic part Case study #2: A posteriori articulation for Pareto optimal solutions. Fig. 3: (left) Results of ten constrained op- timization runs with different colors used for each solution trajectory. (right) Pareto front (filled circles) for the obtained solutions with best trade-off point (blue circle). (a) producer D-1CH 0 500 1000 1500 2000 2500 3000 0 0.5 1 1.5 2 2.5 3 3.5 4 x 10 4 days CO 2 production rate true initial α = 0.0 (seism) best trade-off α = 1.0 (prod) (b) producer D-2H 0 500 1000 1500 2000 2500 3000 0 2 4 6 8 10 x 10 4 days CO 2 production rate (c) producer B-1H 0 500 1000 1500 2000 2500 3000 0 2 4 6 8 10 x 10 4 days CO 2 production rate (d) noise 2% production & 50% seismic 0 1 2 3 4 5 6 7 8 9 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 years saturation fit initial α = 0.0 (seism) best trade-off α = 1.0 (prod) Fig. 4: (a-c) Predictions for CO 2 production rates for three wells with 2% noise in production and 50% in seismic data. (d) Error in predicted saturation for the same noise in data. C ASE S TUDY #2: 3D M ODEL E-4AH permeability E-4H F-1H F-2H F-4H D-4AH E-3BH E-3H E-3CH E-3AH E-1H E-2H E-2AH F-3H D-3H B-4BH D-3AH B-4AH B-3H D-3BH C-4AH C-1H D-4H B-1H C-4H B-4H D-1CH B-2H B-1BH B-1AH K-3H D-2H B-4DH D-1H C-3H C-2H 0 250 500 750 1000 1250 1500 1750 2000 2250 2500 Geometry: mapped from corner-point grid of Norne model, 49 × 120 × 66, 45,470 active cells Geology: sampled from original model Uncertain parameters: permeability Parameterization: PCA on 1000 realizations Model: compositional (see 2D model), well lo- cation/schedule from original Norne model History: 3 years, rates & seismic with noise F ORECASTING P ARETO F RONTS A posteriori articulation treatment for α ∈ [0, 1] via constrained optimization: • add α to the set of optimization variables • define nonlinear constraints s J 0 prod J 0 seism ≤ ( J 0 prod + ) -J prod ( J 0 seism + ) -J seism ≤ S J 0 prod J 0 seism J min 1 J ∗ 1 J 0 1 J min 2 J ∗ 2 J 0 2 α =0 α =1 initial
Transcript of Matching for Noisy Data of Multiple Types Gradient-based...
Gradient-based Pareto Optimal HistoryMatching for Noisy Data of Multiple TypesOleg Volkov (Stanford University), Vladislav Bukshtynov (Florida Tech),Louis J. Durlofsky, Khalid Aziz (Stanford University)
ABSTRACTThe advantages of the simultaneous integra-tion of production and time-lapse seismic datafor history matching (HM) have been demon-strated in multiple studies. Production dataprovide accurate observations at specific spa-tial locations (wells), while seismic data en-able global, though filtered/noisy, estimates ofstate variables. In this work we present an ef-ficient computational tool for bi-objective HM,in which data misfits for both production andseismic measurements are minimized using anadjoint-gradient approach. This enables us toobtain a set of Pareto optimal solutions definingthe optimal trade-off between production andseismic data conflicting misfits. The impact ofnoise structure and noise level on Pareto opti-mal solutions is investigated in detail. We dis-cuss the existence of the “best” trade-off solu-tion, or least-conflicting posterior model, whichcorresponds to the history matched model thatis expected to provide least-conflicting forecastof future reservoir performance. The overallframework is successfully applied in 2D and 3Dcompositional simulation problems.
PARETO OPTIMAL HMReservoir PDE (states x, model parameters u):
g(x,u) = 0, x(t0) = x0.
Multiobjective function to be minimized:
Jmo =α
J ∗prod
· JprodJ 0prod
+1− αJ ∗seism
· JseismJ 0seism
, α ∈ [0, 1],
production data (well rates q̃j,p)
Jprod(x,u) =Nobs∑k=1
Nqwell∑
j=1
Np∑p=1
Cj,p (qj,p − q̃j,p)2 ,
time-lapse seismic data (phase saturation s̃j,p)
Jseism(x,u) =Nobs∑k=1
Nblock∑j=1
Np∑p=1
Cp (sj,p − s̃j,p)2 .
CASE STUDY #1: 2D MODEL
1 2
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11 12 13
10 20 30 40
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9 Geometry: uniformgrid, 45× 45 blocksGeology: in SGeMS,spherical variogram& target histogramUncertain parame-ters: permeability
Parameterization: PCA on 1000 realizationsModel: compositional (6 components), 4 injec-tors + 9 producers (BHP controlled)History: 200 days, rates & seismic with noiseFluid components: same for 2D/3D models
Component CO2 C1 C2 C3 C4 C10
Initial (%) 1 20 20 15 10 34Injection (%) 100 – – – – –
ACKNOWLEDGMENTSStanford University Reservoir Simulation Research (SUPRI-B), Smart Fields Consortia, Tapan Mukerji (ERE, Stanford)
BI-OBJECTIVE HM RESULTS
10−3
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10−1
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10−1
production part
seis
mic
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0%
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noise: 5% prod + 0% seismnoise: 5% prod + 30% seismnoise: 2% prod + 0% seismnoise: 2% prod + 15% seismnoise: 2% prod + 30% seismnoise: 2% prod + 50% seism
Noise in production and seismic data by stan-dard deviations of measurement error:
σq =
{σq,min, γq · qp < σq,min,
γq · qp, γq · qp ≥ σq,min,
σs = γs(1− φ)(1− strueg ).
Fig. 1: Pareto optimal solutions for different levelsof noise in production and seismic data (2D model).
Case study #1Predictions using HM models:• α = 0: only seismic data,• α = 0.5,• α = 1: only production data,• α = 0.826: best trade-off so-
lution determined at a pointwith the highest curvature
Fig. 2: (a-f) Predictions for CO2
injection (a, b) and production(c–f) rates with 2% noise inproduction and 50% in seismicdata. (g-i) Error in predicted sat-uration ‖sg − strueg ‖L2
/‖strueg ‖L2
with (g) no noise, (h) 5% noise inproduction data and (i) 2% noisein production and 50% in seismicdata.
(a) well #2 (injector)
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n fit
initialα = 0.0 (seism)α = 0.5α = 1.0 (prod)
(b) well #3 (injector)
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initialα = 0.0 (seism)α = 0.5α = 1.0 (prod)best trade−off
(c) well #5 (producer)
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trueinitialα = 0.0 (seism)α = 0.5α = 1.0 (prod)best trade−off
(i) noise 2% production & 50% seismic
0 100 200 300 400 500 6000
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initialα = 0.0 (seism)α = 0.5α = 1.0 (prod)best trade−off
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Case study #2: A posteriori articulationfor Pareto optimal solutions.
Fig. 3: (left) Results of ten constrained op-timization runs with different colors used foreach solution trajectory. (right) Pareto front(filled circles) for the obtained solutions withbest trade-off point (blue circle).
(a) producer D-1CH
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Fig. 4: (a-c) Predictions for CO2 production rates for three wells with 2% noise in production and 50% inseismic data. (d) Error in predicted saturation for the same noise in data.
CASE STUDY #2: 3D MODEL
E-4AH
permeability
E-4H
F-1H
F-2H
F-4H
D-4AHE-3BH
E-3HE-3CH
E-3AH
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E-2HE-2AH
F-3H
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D-3AH
B-4AHB-3H
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C-4AHC-1HD-4H
B-1HC-4H B-4H
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B-1BHB-1AHK-3H
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C-2H
0 250 500 750 1000 1250 1500 1750 2000 2250 2500
Geometry: mapped from corner-point grid ofNorne model, 49× 120× 66, 45,470 active cellsGeology: sampled from original modelUncertain parameters: permeabilityParameterization: PCA on 1000 realizationsModel: compositional (see 2D model), well lo-cation/schedule from original Norne modelHistory: 3 years, rates & seismic with noise
FORECASTING PARETO FRONTSA posteriori articulation treatment for α ∈ [0, 1]via constrained optimization:
• add α to the set of optimization variables
• define nonlinear constraints
sJ 0prod
J 0seism
≤(J 0prod + ε
)− Jprod(
J 0seism + ε
)− Jseism
≤ SJ 0prod
J 0seism
Jmin1 J ∗
1 J 01
Jmin2
J ∗2
J 02
α = 0
α = 1
initial
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