smart fields consortium: optimization of oil field ... fields consortium: optimization of oil field...

NTNU 2014

Smart Fields Consortium: Optimization of oil field

development and operations Oleg Volkov

NTNU, Sept 18, 2014

NTNU 2014

Managed by the Department of Energy Resources Engineering

Smart Fields Consortium: Baker Hughes, BG Group, BP, Chevron, CMG, ConocoPhillips, EcoPetrol, ENI, IBM, NTNU, Petrobras, Saudi Aramco, Shell

Goal: develop efficient software tools for the optimization of oil field development and operations

http://smartfields.stanford.edu

Smart Fields Consortium

Sept 18 Optimization of oil field development and operations 2

http://smartfields.stanford.edu/

NTNU 2014

Smart Fields


• Decision loop consists of the production system, sensors, data aggregation, planning and control

• Optimization techniques could be deployed at any stage of the development of a field

Production System

Data

Update Detailed Model

Update Reduced Model

Optimization

Controls

Optimization

Gradient Framework

Geological Model

Controls

Production system

Production Data

Seismic Data

Field Development

NTNU 2014

Optimization of field operation


• Production optimization • Optimal well control • Few continuous and

discrete valued optimization variables

• Optimization techniques: • Reduced-order models • Proxies • Adjoint-gradient • Data mining from downhole

gauges • Streamline models • Approximate dynamic

programming • Carbon Dioxide Sequestration

Production System

Data



Optimization

Controls

Optimization

Gradient Framework

Geological Model

Controls

Production system

Production Data

NTNU 2014

History matching


Data sources scarce in time and space: historical production data, seismic data etc.

Inverse modeling Numerous

continuous and discrete valued unknown variables

Geological uncertainty: quantification and propagation

Techniques: tomographic inversion PCA+adjoint gradient and PSO

Production System

Data



Optimization

Controls

Optimization

Gradient Framework

Geological Model

Controls

Production system

Production Data

Seismic Data

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Field development


• Well placement • Well pattern optimization • Evolutionary algorithms • Mesh adaptive direct search • Pattern search

• Drilling decision support : well and well completion design • Bayesian networks

• Discrete and

continuous variables

• Geological uncertainty

Production System

Data



Optimization

Controls

Optimization

Gradient Framework

Geological Model

Controls

Production system

Production Data

Field Development

NTNU 2014

Survey from E&P companies, 136 participants Data management is stated to be a challenge by 90%

Respondents admitted that they would like to know more about 25% - how to use the data which is collected for internal

decision making and external evaluation 23% - how to establish a uniform method of data

assessment 21% - how to create a data environment in which the full

potential of big data can be harnessed

Industrial perspective


NTNU 2014

Louis Durlofsky, Khalid Aziz, Hamdi Tchelepi, Tapan Mukerji, Marco Thiele

Oleg Volkov, Vladislav Bukshtynov adjoint-based gradient framework for reservoir simulator, constrained production optimization, history matching, re-parameterization, closed-loop

Elnur Aliyev efficient field development optimization under uncertainty using upscaled models

Mehrdad Gharib Shirangi closed-loop field development optimization Hai X. Vo geological parameterization for history matching complex

models Matthieu Rousset optimization-based framework for geological

scenario determination using parameterized training images Sumeet Trehan, Rui Jiang reduced order models, TPWL, TPWQ

Contributors


NTNU 2014

• General Purpose Research Simulator • Automatic Differentiation-based GPRS • Optimization toolkit for AD-GPRS • SNOPT (NLP solver) • PSO-MADS global search hybrid algorithm • Field development optimization framework • SGeMS

Computational tools


NTNU 2014

Efficient field development optimization under uncertainty using upscaled models Elnur Aliyev & Louis J. Durlofsky


NTNU 2014 Sept 18 Optimization of oil field development and operations 11

• Field development optimization methods can be computationally costly

• Optimization under uncertainty entails evaluating performance over multiple realizations (very costly)

• Flow-based surrogate models are attractive for reducing computation

• Existing reduced-order models (POD-TPWL) still only applicable for fixed well locations

Motivation

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Upscaling methodology


Global fine grid: 15x15 Global coarse grid: 5x5

x

x

x

x

x x

x

x

Well locations defined on fine grid

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Computation of upscaled terms


x

x

x

x

Global fine grid

j- j j+

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Iterative upscaling procedure


• Initial implementation by Matthieu Rousset

• Replace negative/anomalous Tj* with analytical (geometric) average

• Solve coarse model with updated Tj*

• Use new pressure values to compute Tj*

• Replace negative/anomalous (Tj* )ν with (Tj* )ν-1

• Iterate (ν ν +1)

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Field development optimization


• Implementation by Obi Isebor • PSO-MADS global search hybrid algorithm • Components:

– Well placement (discrete/categorical variables)

– Production optimization (continuous variables)

• Nonlinear constraint handling using filter method

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Optimization with model refinement


10x10 2 sec

Level 1

Simulation time:

50x50 20 sec

Level 3

100x100 120 sec

Level 4

25x25 8 sec

Level 2

Early sub-problems faster to evaluate; later sub-problems converge quickly because initial guess is close to optimum

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3D Example


• Optimize vertical well location and completion interval in 3D model

• 30 x 30 x 6 grid, BHP-controlled wells – 4 producers (fixed at 1500 psi) – 1 injector (fixed at 6500 psi)

• 4 variables per well • 20 decision variables • Economic data:

– Oil price: $100/STB – Water inj: $5/STB – Water prod: $10/STB – Well cost: $5 million/well

NTNU 2014

3D Example: Optimization performance


10x10x1 15x15x2 30x30x3 30x30x6 Total time, hours

Upscaling & simulation time, sec

6 9 39 111

# of function evaluations (FEs) 1741 926 461 192

Run 1 $B

Run 2 $B

Run 3 $B

Average NPV, $B STD, $B

Time spent, hours

Conventional method 1.21 1.16 1.20 1.19 0.025 3.4

Model refinement 1.21 1.19 1.20 1.20 0.010 0.49

Conventional method requires 3320 objective function evaluations

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Closed-loop field development optimization Mehrdad Gharib Shirangi & Louis J. Durlofsky


NTNU 2014

Motivation


Drill New

Well

Optimize Well Placement

Reservoir Data

Model Updating

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Closed-loop


𝑡1

Optimization

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Closed loop


𝑡1

𝑡2

Production from Well 1

Optimization

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Closed loop


𝑡1

𝑡2

History Matching

Optimization


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Closed loop


𝑡1

𝑡2

History Matching

Optimization Optimization


NTNU 2014

Closed loop


𝑡1

𝑡2


Prod / Inj from Wells 1 & 2

𝑡3 History Matching


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Closed loop


𝑡1

𝑡2



𝑡3

History Matching

History Matching


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Closed loop


𝑡1

𝑡2



𝑡3

History Matching

Optimization

History Matching


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Closed loop


𝑡1

𝑡2…𝑡6

𝑡7

History Matching


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Optimization problem


Objective function for field development optimization: 𝑁𝑁𝑁 = 𝐽 = 𝑝𝑜𝑄𝑜 − 𝑐𝑤𝑤𝑄𝑤𝑤 − 𝑐𝑤𝑤𝑄𝑤𝑤 − ∑𝑐𝑤𝑤𝑤𝑤

𝐽 = 𝐽 𝑢, 𝑚𝑗

𝑤 𝑢: vector of decision parameters (number of wells, well types,

controls, locations, drilling sequence)

𝑚𝑗𝑤: 𝑗-th realization updated at time 𝑡𝑤

Robust optimization:

𝐽 ̅ =1

𝑁𝑤� 𝐽 𝑢, 𝑚𝑗

𝑤𝑁𝑒

𝑗=1

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Optimization problem


𝐽 ̅ =1

𝑁𝑤 � 𝐽 𝑢, 𝑚𝑗

𝑤 𝑁𝑒

𝑗 =1

𝑀𝑤 = 𝑚1

𝑤 , 𝑚2𝑤 … 𝑚𝑁𝑒

𝑤 : is the set of realizations updated at 𝑡𝑤

𝐽 ̅ = 𝐽 ̅ 𝑢, 𝑀𝑤

Optimal solution (at 𝑡𝑤) : 𝒖𝒊 = argma𝑥 𝐽 ̅ 𝑢, 𝑀𝑤 , using PSO-MADS (Isebor et al 2013)

Use 𝑢𝑤−1 as initial guess when optimizing at time 𝑡𝑤

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History matching & Bayesian Framework


Minimize

𝑆 𝑚 =12

𝑚 − 𝑚�𝑤𝑝𝑤𝑜𝑝𝑇

𝐶𝑀−1 𝑚 − 𝑚�𝑤𝑝𝑤𝑜𝑝

+ 12

𝑔 𝑚 − 𝑑𝑜𝑜𝑜𝑇𝐶𝐷

−1(𝑔 𝑚 − 𝑑𝑜𝑜𝑜)

𝑑𝑜𝑜𝑜: observed data (vector), BHP, phase rates 𝑔 𝑚 : predicted data (vector), BHP, phase rates 𝐶𝐷: (diagonal) covariance matrix for measurement errors

Minimizing 𝑆(𝑚) gives the maximum a posteriori (MAP) estimate

Model mismatch term (prior)

Data mismatch term (likelihood)

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History matching and hard data


Minimize

𝑆 𝑚 =12

𝑚 − 𝑚�𝑤𝑝𝑤𝑜𝑝𝑇

𝐶𝑀−1 𝑚 − 𝑚�𝑤𝑝𝑤𝑜𝑝

+12

𝑔 𝑚 − 𝑑𝑜𝑜𝑜𝑤 𝑇

𝐶𝐷,𝑤−1 𝑔 𝑚 − 𝑑𝑜𝑜𝑜

𝑤

+12

𝑚ℎ − 𝑑𝑜𝑜𝑜ℎ 𝑇

𝐶𝐷,ℎ−1 𝑚ℎ − 𝑑𝑜𝑜𝑜

ℎ

𝑑𝑜𝑜𝑜ℎ : vector of observed model parameters (hard data)

𝑚ℎ: current estimate for observed model parameters 𝐶𝐷,ℎ : (diagonal) covariance matrix for measurement errors

Model mismatch term

Production data

Hard data

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Randomized maximum likelihood


Generate 𝑛 samples from the prior pdf 𝒎𝒖𝒖~𝑁 𝑚𝑤𝑝𝑤𝑜𝑝 , 𝐶𝑀

Generate 𝑛 samples as

𝒅𝒖𝒖~𝑁 𝑑𝑜𝑜𝑜, 𝐶𝐷

Minimize 𝑛 objective functions to generate 𝑁 posterior samples using L-BFGS

𝑆 𝑚 =12

𝑚 − 𝒎𝒖𝒖𝑇𝐶𝑀

−1 𝑚 − 𝒎𝒖𝒖

+12

𝑔 𝑚 − 𝒅𝒖𝒖𝑇𝐶𝐷

−1 𝑔 𝑚 − 𝒅𝒖𝒖

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Subset of representative realizations


- History match 50 realizations - Discard realizations with 𝑆𝑁 > 10

- Run realizations with 𝑢𝑤−1 - Choose 10 representative realizations

- Optimize - Drill the next well - Collect reservoir data

0 20 40-5-4-3-2-10x 108

ranked realization index

- NP

V

Ne=10

0 10 20 30 40 50

110

1001000

10000

simulation runs

SN

(m) (Similar to

Elnur Aliyev’s treatment)

NTNU 2014

3D example: 30 x 30 x 5


Uncertain model parameters: ln(𝑘) Drill 6 wells : 3 horizontal producers, 3 vertical injectors History match and optimize over 𝑁𝑤 = 6 realizations

Layer 1

X

Y

5 10 15 20 25 30

5

10

15

20

25

30

2

3

4

5

6

7

parameter value

well cost $ 25 million oil price $ 90 / bbl produced water $ 10 / bbl injected water $ 10 / bbl drilling lag-time 210 Days reservoir Life 2000 Days perforation cost $ 1 million /gb

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Optimal NPV versus Update Steps of CLFD (𝑁𝑤 = 6)


𝐽 𝑢𝑤 , 𝑀𝑤 : Optimal E[NPV] updated at 𝑡𝑤

𝐽(𝑢𝑤 , 𝑚𝑡𝑝𝑡𝑤 ): NPV for the true model (run the true model with 𝑢𝑤)

0 210 420 630 840 1050 12608.5

9

9.5

10

10.5x 108

Time (Days)

NP

V ($

)

J(ui , Mi )J(ui , mtrue )

Deterministic

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Geological parameterization for history matching complex models Hai X. Vo & Louis Durlofsky


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Motivation


PCA is simple and linear but gives “Gaussian-looking” models and histograms

Intent with O-PCA is to modify PCA procedure to better represent geologically-complex models

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History matching problem


𝒎𝒊𝒎𝒚

𝑺 = � 𝒅𝒐𝒐𝒐𝒎 − 𝒅𝒐𝒊𝒎

𝒎 (𝒚) 𝟐𝑵−𝟏

𝒎=𝟎

+ 𝒓𝒓𝒓𝒖𝒓𝒓𝒓𝒊𝒓𝒓𝒓𝒊𝒐𝒎 𝒓𝒓𝒓𝒎

𝒚 = 𝒌𝟏, 𝒌𝟐, … . , 𝒌𝑵𝒖

𝑻

Idea is to find a geological model 𝒚 such that prediction matches production and honors prior information

NTNU 2014

Challenges of history matching


Real models can have millions of grid blocks

Updating permeabilities for all blocks independently is expensive and may not maintain geology

Useful to have algorithms that parameterize reservoir model as 𝒚 𝝃 and maintain geology

Can then write history matching problem as:

𝒎𝒊𝒎𝝃

𝑺 = � 𝒅𝒐𝒐𝒐𝒎 − 𝒅𝒐𝒊𝒎

𝒎 (𝒚(𝝃)) 𝟐𝑵−𝟏

𝒎=𝟎

+ 𝒓𝒓𝒓𝒖𝒓𝒓𝒓𝒊𝒓𝒓𝒓𝒊𝒐𝒎

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Constructing PCA representation


Run geostatistical algorithm (SGeMS) to create a set of N realizations: 𝒚𝟏, 𝒚𝟐, 𝒚𝑵

Construct

Conceptually, define covariance matrix as

Eigen-decomposition of 𝑪 to give: (in practice use SVD of 𝒀)

K-L (PCA) representation

𝑪 =𝟏𝑵

𝒀𝒀𝑻 𝑼𝜦𝑼𝑻= C

𝒚𝒎𝒓𝒏 = 𝑼𝒓𝜦𝒓𝟏/𝟐𝝃 + 𝒚�

= 𝚽𝒓 𝝃 + 𝒚� (l << N)

𝒀 = 𝒚𝟏 − 𝒚�, 𝒚𝟐 −𝒚�, … . , 𝒚𝑵 −𝒚�

= +

𝒚𝒎𝒓𝒏 = 𝚽𝒓 𝝃 + 𝒚�

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History matching using PCA


Maximum A Posteriori (MAP) estimate

Single model

Set

Subspace Randomized Maximum Likelihood (RML):

Multiple models for uncertainty quantification

Set

𝑺 = 𝟏𝟐

𝑷𝒐𝒊𝒎 𝝃 − 𝑷𝒐𝒐𝒐∗ 𝑻 𝑪𝑫

−𝟏 𝑷𝒐𝒊𝒎 𝝃 − 𝑷𝒐𝒐𝒐∗ + 𝟏

𝟐𝝃 − 𝝃∗ 𝑻(𝝃 − 𝝃∗)

𝝃∗ = 𝝃𝒖𝒖

𝝃∗ = 𝝃� = 𝟎

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Gaussian fields


?

SGeMS realization 𝒚𝒎𝒓𝒏 = 𝚽𝒓 𝝃 + 𝒚� PCA :

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Binary facies model


SGeMS PCA

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Three-facies model


SGeMS PCA

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Bimodal models


SGeMS PCA

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Optimization-based PCA


Standard PCA: 𝒚𝒎𝒓𝒏 = 𝚽𝒓𝝃 + 𝒚�

𝒚𝒎𝒓𝒏 = 𝐚𝐚𝐚𝐚𝐚𝐚𝒓

𝚽𝒓𝝃 + 𝒚� − 𝒓 𝟐𝟐

Formulate PCA as an optimization problem:

Apply regularization + bound constraints:

𝒚𝒎𝒓𝒏 = 𝐚𝐚𝐚𝐚𝐚𝐚𝒓

𝚽𝒓𝝃 + 𝒚� − 𝒓 𝟐𝟐 + γ𝑹

𝒓 ∈ [𝒓𝒓, 𝒓𝒖]

𝑹 depends on geological model, e.g. binary 𝑹 = 𝒓𝑻(𝟏 − 𝒓)

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History matching and O-PCA


Need for gradient-based history matching

Construct using:

O-PCA gives analytically

𝒅𝑺𝒅/𝒅𝝃

𝒅𝑺𝒅

𝒅𝝃=

𝒅𝑺𝒅

𝒅𝒌𝒅𝒌𝒅𝒚

𝒅𝒚𝒅𝝃

from O-PCA

from simulator

𝒌 = 𝒇 𝒚

𝒅𝒚/𝒅𝝃

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Example: three-facies system


Oil-water with AD-GPRS

2D model, 100 x 100, 𝒓 = 100

2 injectors, q=1500 m3/d, 2000 m3/d

6 producers, BHP = 100 bar

kchannel = 2000 md, klevee = 200 md, kmud = 20 md

Data: injection pressures and production rates for 1200 days

SGeMS realizations (and O-PCA models) conditioned to hard data

Hard data

SGeMS realization

Producer Injector

20 40 60 80 100

20

40

60

80

100

0

0.5

1

1.5

2

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MAP estimate


Channel Levee

Shale

I1

I2P1P2

P3P4

P5

P6

20 40 60 80 100

20

40

60

80

100

0

0.5

1

1.5

History matched

𝑺 = 𝟏𝟐

𝑷𝒐𝒊𝒎 𝝃 − 𝑷𝒐𝒐𝒐∗ 𝑻 𝑪𝑫

−𝟏 𝑷𝒐𝒊𝒎 𝝃 − 𝑷𝒐𝒐𝒐∗ + 𝟏

𝟐𝝃𝑻𝝃

Initial guess

True model

I1

I2P1P2

P3P4

P5

P6

20 40 60 80 100

20

40

60

80

100

0

0.5

1

1.5

2

I1

I2P1P2

P3P4

P5

P6

20 40 60 80 100

20

40

60

80

100

0

0.5

1

1.5

2

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Field rates


0 1000 2000 3000 40000

500

1000

1500

2000

2500

3000

Days

TrueInitialHM

Initial

True HM

period

HM

Fiel

d W

ater

Rat

e, m

3 /d

Days 0 1000 2000 3000 4000

0

1000

2000

3000

4000

Days

TrueInitialHMInitial

True HM

HM period

Fiel

d O

il R

ate,

m3 /d

Days

Field Oil Rates Field Water Rates

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Example: prediction of uncertainty


Oil-water with AD-GPRS

2D model, 60 x 60, 𝒓 = 100

2 injectors, q = 1500 m3/d

5 producers, BHP = 150 bar

ksand = 2000 md, kmud = 20 md

Data: injection pressures and production rates for 1200 days

SGeMS realizations (and O-PCA models) conditioned to hard data

Hard data

SGeMS realization

Producer Injector

20 40 60

10

20

30

40

50

60

0

0.2

0.4

0.6

0.8

1

NTNU 2014

Prior and posterior field oil rates


Prior models conditioned only to hard data at wells, posterior models history matched to flow data

True

HM period

Fiel

d O

il R

ate,

m3 /d

Days

Prior Models

True

HM period

Fiel

d O

il R

ate,

m3 /d

Days

Posterior Models

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Optimization toolkit of AD-GPRS

Oleg Volkov & Vladislav Bukshtynov


NTNU 2014

Optimization w.r.t. continuous-valued parameters as opposed to discrete-valued categorical parameters

Production Optimization e.g. BHP, phase rates and completion transmissibility factors

History Matching e.g. permeabilities

Validation of the gradients and AD derivatives Implementation into state-of-the-art nonlinear

programming (NLP) solvers Input data consistent with Eclipse

Features


NTNU 2014

Input file in an ordinary text file Always starts with SIMULATOR (name of the AD-GPRS

input data file)

Content of keywords OPTDIMS, OPTPARS, OPTCONS, and OPTFUNC is consistent with Eclipse

OPTOPTS and OPTTUNE altered for tuning our optimization algorithm

A new keyword OPTCHCK is introduced for the consistency tests

Compatibility with Eclipse


NTNU 2014

History matching is the process of building one or more sets of reservoir model parameters which account for observed, measured data

Currently parameters are permeability and interblock transmissibility (multipliers)

Sample a posteriori probability density function (pdf) using randomized maximum likelihood (RML) method

Re-parameterization

principal component analysis (PCA)

parameter grouping based on geological knowledge (faults, barriers etc.) or user defined patterns

Capabilities of History Matching


NTNU 2014

Maximizing economic value (cash flow) Cash flow = revenues − technical costs −government take

Reduced to a linear function of production and injection rates

𝑆 = � � 𝑁𝑤𝑞𝑤,𝑤

phases

𝑤

producers

𝑤

+ � � 𝐶𝑤𝑞𝑤,𝑗

phases

𝑤

injectors

𝑗

Discounting = reducing the value of money over time to reflect the return on investment that could have been made by investing the money elsewhere

Discount factor in one year 𝑆𝑑𝑤𝑜𝑑 = 𝑆 × 1 + 𝑟𝑑𝑤𝑜𝑑/100 −1

𝑟𝑑𝑤𝑜𝑑 − discount rate in % per year Net Present Value = discounted cumulative cash surplus

Production optimization problem


𝑁𝑁𝑁 = � 𝑆𝑑𝑤𝑜𝑑 ∆𝑡end of project

𝑡=0

NTNU 2014

Production optimization example


Net Present Value Bounds and nonlinear constraints

Producer well liquid rate < 6000 𝑚3 𝑑𝑑𝑑⁄ Injector well water rate < 12000 𝑚3 𝑑𝑑𝑑⁄ Well water cut < 95% or Field water cut < 95% Injector BHP < 450 Bar Producer BHP > 150 Bar

𝑁𝑁𝑁 = � 11.1

𝑡365�

75 $𝑜𝑜𝑤 𝑞𝑜,𝑤 − 6 $

𝑜𝑜𝑤 (𝑞𝑤,𝑤 + 𝑞𝑤,𝑤) − 1.2 $𝑀𝑜𝑑𝑀 𝑞𝑔,𝑤 ∆𝑡

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History matching problem


Objective function (least square mismatch)

model parameters, uncertainty

, covariance matrix may be approximated by

sample covariance built on prior knowledge of geological features

analytic covariance based on analysis of available variograms

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OPTDIMS – numbers of iterations and number of simulations as a termination criteria

OPTFUNC – definition of the objective function FIELD mnemonics only Production optimization: FOPT, FWPT, FWIT, FGPT, FGIT

e.g. History matching: HMOP, HMGP, HMWP, HMWI, HMGI, HMPP, HMPI e.g.

Main keywords in OptADGPRS


OPTFUNC FOPT FIELD 471.73580778240785 0.1 / FWIT FIELD -37.738864622592629 0.1 / FWGT FIELD -0.0423776000657863 0.1 //

OPTFUNC HMOP FIELD 1.0 / HMGP FIELD 1.0 / HMWP FIELD 1.0 / HMWI FIELD 1.0 / HMGI FIELD 1.0 //

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OPTPARS – definition of the control variables WBHP, WOPR, WWPR, WGPR, WGIR, WWIR – well control CTRF – well transmissibility factor control e.g.

PERM – control using permeability multipliers for history matching e.g.

LOGPARS – log-scale parameterization

Control parameters in OptADGPRS


OPTPARS WBHP F-1H 250 450 / WBHP E-3H 150 300 2 / WBHP E-3AH 150 300 5 6 / CWIN F-1H 0 0 10 0 2 / CWIN F-3H 1:4 0 10 / CWIN E-2H * 0 10 //

OPTPARS PERM * 0 1e+3 / LOGPARS //

NTNU 2014

Given desired goal and inequality constraints involving 𝑥 and/or 𝑢 find optimal 𝑢opt

where

Unified optimization problem


find optimum of 𝐽(𝑥, 𝑢)subject to 𝑔 𝑥, 𝑢 = 0, ℎ(𝑥, 𝑢) ≤ 0

𝑥 = state variables (pressure, saturations, etc.) 𝑔 = state equations (mass balance equations) 𝐽 = objective (Net Present Value, mismatch of observables) 𝑢 = optimization variables (well controls, modeling parameters) ℎ = nonlinear constraints (phase rates, gas-oil ratio, etc.)

NTNU 2014

Define augmented Lagrangian

where 𝜆 is the Lagrange multiplier for 𝑔

First order optimality conditions require zeros of the derivatives of Lagrangian with respect to 𝜆, 𝑥, 𝑢

Karush-Kuhn-Tucker optimality conditions


ℒ 𝑥, 𝑢, 𝜆 = 𝐽 𝑥, 𝑢 + 𝜆𝑇𝑔(𝑥, 𝑢)

𝑔 𝑥, 𝑢 = 0 state equation 𝑔𝑥

𝑇 𝑥, 𝑢 𝜆 = −𝐽𝑥(𝑥, 𝑢) adjoint equation

𝑑ℒ = 𝐽𝑡 𝑥, 𝑢 + 𝑔𝑡𝑇 𝑥, 𝑢 𝜆 ∙ 𝛿𝑢

𝛻𝑡𝐽

min𝛿𝑡

𝛻𝑡𝐽 ∙ 𝛿𝑢

NTNU 2014

Directional derivative of objective function 𝐽 for a perturbation 𝛿𝑢

Riesz identity

OPTCHCK Number of consistency checks of the adjoint gradient Frequency of the consistency checks during the optimization Perturbation value 𝜏 Tolerance for print Type of the finite difference scheme used in the consistency check:

forward, backward, central Message CONSISTENCY_TEST_FAILED in the .sim.log file

Gradient consistency check


𝑑𝐽 𝑥, 𝑢; 𝛿𝑢 = lim𝜏⟶0

𝐽 𝑥 𝑢 + 𝜏𝛿𝑢 , 𝑢 + 𝜏𝛿𝑢 − 𝐽(𝑥, 𝑢)𝜏

𝑑𝐽 𝑥, 𝑢; 𝛿𝑢 = 𝑔𝑡𝑇 𝑥, 𝑢 𝜆𝑇 + 𝐽𝑡 𝑥, 𝑢 , 𝛿𝑢

NTNU 2014

Optimality criteria based on Karush-Kuhn-Tucker conditions

OPTTUNE Scaling factor for the objective function

reducing objective function to ∼ 1 improves convergence for certain problems

Convergence tolerance tolerance for the KKT overall violation (in SNOPT normalized by the dual variables)

Constraint tolerance tolerance for the constraint violation (in SNOPT normalized by the control values)

Optimality tolerance tolerance on the complementarity and dual infeasibility

Tuning parameters in OptADGPRS


NTNU 2014

ADETL: computation of Jacobian & partial derivatives of objective functional with respect to states and control variables using saved states variables (hdf5)

GMRES + AMG/SAMG/PARDISO - CPR preconditioner for adjoint system (𝑔𝑥

𝑇 is transposed Jacobian)

Bounds and constraints nonlinear programming solver

Implementation in AD-GPRS


𝑔𝑥𝑇 𝑥, 𝑢 𝜆 = −𝐽𝑥(𝑥, 𝑢)

min𝛿𝑡

𝑔𝑡𝑇 𝑥, 𝑢 𝜆 + 𝐽𝑡(𝑥, 𝑢) ∙ 𝛿𝑢

NTNU 2014

Linear solver for adjoint system

Schur-complement for perturbed residual: 𝑅𝑅 → 𝑅𝑅 − 𝑅𝐹 𝐹𝐹−1𝐹𝑅, 𝑅𝐹 → 0

for adjoint system: 𝐽𝑥𝑅 → 𝐽𝑥𝑅 − 𝐹𝑅𝑇𝐹𝐹−𝑇𝐽𝑥𝐹, 𝐹𝑅𝑇 → 0

GMRES + CPR preconditioner for 𝑅𝑅𝑇 (Drosos Kourounis)

Linear solver


𝑔𝑥𝑇 𝑥, 𝑢 𝜆 = −𝐽𝑥(𝑥, 𝑢)

𝑅𝑅𝑇

𝑅𝐹𝑇

𝐹𝑅𝑇

𝐹𝐹𝑇

𝜆𝑅𝑇

𝜆𝐹𝑇

= −𝐽𝑥𝑅

−𝐽𝑥𝐹

NTNU 2014

OPTOPTS – solver options Optimizer: SNOPT, IPOPT Gradient evaluation method: DISCRETE_(AD,FD),

FINITE_DIFFERENCES Linear solver:

direct: SUPERLU TRANSPOSE, PARDISO TRANSPOSE iterative + CPR preconditioner:

Krylov methods: Generalized Minimal Residual Method (GMRES), Conjugate Gradient method (CGS), or BIConjugate Gradient method (BICGSTAB)

preconditioner left (@LEFT) or right (@RIGHT) solver for pressure system (CPR_PARDISO, CPR_AMG, CPR_SAMG)

Maximum number of Krylov iterations Tolerance of the relative residual

Solvers in OptADGPRS


NTNU 2014

Nonlinear inequalities ℎ(𝑥, 𝑢) ≤ 0

NLP solvers efficient treatment for local infeasibility require the gradient of ℎ w.r.t. 𝑢

Adjoint-based gradient of nonlinear constraints computed together with objective gradient

What about non-differentiable functionals?

e.g. ℎ 𝑥, 𝑢 ≤ 0, ∀𝑡 ⟹ max𝑡

ℎ(𝑥, 𝑢) ≤ 0

Implementation of constraints


𝑔𝑥𝑇 𝑥, 𝑢 𝜇𝑇 = −ℎ𝑥(𝑥, 𝑢)

𝛻𝑡ℎ = 𝑔𝑡𝑇 𝑥, 𝑢 𝜇 + ℎ𝑡(𝑥, 𝑢)

NTNU 2014

Softmax Function


Non-differentiable constraints at discrete time 𝑡𝑤

Solution: smooth maximum function + AD

max𝑤

𝑞 𝑥𝑤 , 𝑢𝑤 ≤ 𝑄

max 𝑑, 𝑏 =12

𝑑 + 𝑏 + (𝑑 − 𝑏)2+𝜀2

4 4.5 5 5.5 64

4.5

5

5.5

6

softm

ax(a

,b)

a4 4.5 5 5.5 6

-0.5

0

0.5

1

1.5

deriv

ativ

e w

.r.t.

aa

softmax(a,b), ε=1softmax(a,b), ε=0.1a b=10-a

Recursive calculation for multiple 𝑞𝑤

NTNU 2014

OPTCONS – constraints to be satisfied during optimization {F,W}OPR, {F,W}WPR, {F,W}GPR, {F,W}WIR, {F,W}GIR, {F,W}LPR – field/well phase rates

{F,W}GOR – field/well gas-oil ratio {F,W}WCT – field/well water cut

e.g. for the well P1 max

𝑡 𝑞𝑔,𝑃1 ≤ 2000, ∀𝑡 ∈ 2,3,4 input regions

for three injectors I1, I2, I3 min𝑡

min(𝑞𝑤,𝐼1, 𝑞𝑤,𝐼2, 𝑞𝑤,𝐼3) ≥ 500

for field production max𝑡

𝑞𝑔,𝑃1 + 𝑞𝑔,𝑃2 + 𝑞𝑔,𝑃3 ≤ 1.0E+6

Constraints


OPTCONS WGPR P1 < 2000.0 2 4 / WWIR * > 500.0 / FGPR FIELD < 1.0E+6 //

NTNU 2014

Evaluate objective and constraint functionals

Evaluate adjoint based gradients

Non-Linear Programming (NLP) framework

Initialize optimization

problem

Optimization Module

Optimization Parameters

Gradient Evaluator

AD-GPRS Forward

simulator

memory alloc

bounds

Control Center

call forward solver, read output data

Objectives and

constraints

Keywords

Optimization

assemble functional

assemble gradient

Input data

NLP solver

termination criteria

new set of controls

output

Data flow


Transpose CPR solver

call adjoint solver

NTNU 2014

Initialization 1. Through input data of the simulator

2. OPTLOAD – raw data to initialize control variables of the optimization problem.

Internal order of the control variables: it starts with an array of values corresponding to the first valid control parameter in OPTPARS at all control steps, followed by an array of the second parameter at all control steps etc.

Output .sim.log – general log file .opt.pars – objective, constraint, and controls at each evaluation of objective .opt.prnt – SNOPT log file .opt.summ – SNOPT summary file .opt.log – IPOPT log file

Initialization and output


NTNU 2014

Examples and applications

Production optimization with economic limits HM and PCA-based parameterization History matching of geological features

HM with multiple data types Closed-loop field management


NTNU 2014

PO with economic limits


NTNU 2014

Previous work: maximal well flow rate constraints Kourounis et al. (2014) Adjoint formulation and constraint handling for

gradient-based optimization of compositional reservoir flow, Computational Geosciences

Volkov O., Voskov D. (2013) Advanced Strategies of Forward Simulation for Adjoint-Based Optimization, SPE163592

Conclusion: production optimization with well flow rate constraints in optimization does not perform as well as production optimization with constraints in simulation Sequential Quadratic Programming Lumped constraint using smoothed maximum function

Economic limits Well water cut limit Minimal well injection rate limit

Production optimization with economic limits

Optimization of oil field development and operations 78 Sept 18

NTNU 2014

Constraints in simulation Shut-in well

Violation of the limit is verified after each time step. If violated, the well is closed until the end of the simulation.

Stopped well Violation of the limit is verified within each time step. If violated, the well is switched to a zero-rate control, i.e. it has cross flow and can be reopened.

Constraints in optimization One lumped constraint for all wells Individual constraint for each well

Implementation of economic limits


NTNU 2014

Van Essen et al. (2006) Robust waterflooding optimization of multiple geological scenarios, SPE102913-PA

Systematic comparison: simulations with constant time step of 30 days

Test case #1

Optimization of oil field development and operations 80

𝑁𝑁𝑁 = � 11.1

𝑡365�

75 $𝑜𝑜𝑤 𝑞𝑜,𝑤 − 4 $

𝑜𝑜𝑤 (𝑞𝑤,𝑤 + 𝑞𝑤,𝑤) 𝑑𝑡

Sept 18

BHP controlled production (4 wells) and injection (8 wells)

10 control steps of 180 days each ≈ 5 years

Economic limit: well water cut < 95%

NTNU 2014

Maximum water cut ≠ limit 0.9220 (lumped) 0.9421 (individual well) 0.9482 (shut-in well) 0.9243 (stopped well)

Number of simulations 35 (lumped) 20 (individual well) 35 (shut-in well) 33 (stopped well)

Economic limits in SQP


Sequential Quadratic Programming Termination when solution cannot be improved • 20 major iterations, gradient tolerance of 10−3, feasibility region

Large major step limit, derivative-based line search

Sept 18

0 500 1000 15000

0.5

1

1.5

2

2.5

x 108

days

NP

V,$

constraint in optimization (lumped)constraint in optimization (individual well)constraint in simulation (shut-in well)constraint in simulation (stopped well)base

NTNU 2014

Economic limits in MMA and IPM


Method of Moving Asymptotes Nonlinear approximation of the constraints

Interior Point Method Barrier functions for nonlinear constraints

Sept 18

Max water cut ≠ limit Number of simulations

lumped 0.9392 126 individual well 0.9479 63

shut-in well 0.8461 24

stopped well 0.9070 41

Max water cut ≠ limit Number of simulations

lumped 0.9522 91 individual well 0.9127 26

shut-in well 0.9293 1068

stopped well 0.9094 143

0 500 1000 15000

0.5

1

1.5

2

2.5

x 108

days

NP

V,$


0 500 1000 15000

0.5

1

1.5

2

2.5

x 108

days

NP

V,$


NTNU 2014

SQP (solution for well BHP)

MMA (solution for well BHP)

Optimal controls


base

0 900 1800250

260

270

280

time, days

0 900 1800240

250

260

stopped well

0 900 1800250

260

270

280

time, days

0 900 1800240

250

260

shut-in well

0 900 1800250

260

270

280

time, days

0 900 1800240

250

260

individual well

0 900 1800250

260

270

280

time, days

0 900 1800240

250

260

inje

ctor

s

lumped

0 900 1800250

260

270

280

prod

ucer

s

time, days

0 900 1800240

250

260

base

0 900 1800250

260

270

280

time, days

0 900 1800240

250

260

stopped well

0 900 1800250

260

270

280

time, days

0 900 1800240

250

260

shut-in well

0 900 1800160180200220240260280

time, days

0 900 1800160180200220240260280

individual well

0 900 1800250

260

270

280

time, days

0 900 1800240

250

260

inje

ctor

s

lumped

0 900 1800250

260

270

280

prod

ucer

s

time, days

0 900 1800240

250

260

NTNU 2014

Norne Field: offshore Norwegian Sea, operated by Statoil, Norway 42,239 active grid blocks [46 × 112 × 22] Corner Point Grid, faults, and pinch-outs Real field geological and initial simulation data Base case conditioned at Jan 1, 2007 (BHP controls, 31 wells)

Test case #2


red - 𝑆𝑔𝑔𝑜

green - 𝑆𝑜𝑤𝑤

blue - 𝑆𝑤𝑔𝑡𝑤𝑝

Sept 18

NTNU 2014

Production optimization outline


Net Present Value Constraints

Producer well liquid rate < 6000 𝑚3 𝑑𝑑𝑑⁄ Injector well water rate < 12000 𝑚3 𝑑𝑑𝑑⁄ Well water cut < 95% Injector BHP < 450 Bar Producer BHP > 150 Bar

Optimization strategies (control variable: well BHP)

Sequential Quadratic Programming (SNOPT) Method of Moving Asymptotes (NLOPT)

𝑁𝑁𝑁 = � 11.1

𝑡365�

75 $𝑜𝑜𝑤 𝑞𝑜,𝑤 − 6 $

𝑜𝑜𝑤 (𝑞𝑤,𝑤 + 𝑞𝑤,𝑤) − 1.2 $𝑀𝑜𝑑𝑀 𝑞𝑔,𝑤 𝑑𝑡

Sept 18

NTNU 2014

MMA vs. SQP


base field case MMA SQP

'06 '08 '10 '12 '14 '160

1

2

3

4x 10

8

years

wat

er in

ject

ion,

bbl

'06 '08 '10 '12 '14 '160

0.5

1

1.5

2

2.5x 10

8

years

wat

er p

rodu

ctio

n, b

bl

'06 '08 '10 '12 '14 '160

2

4

6

8x 10

7

years

oil p

rodu

ctio

n, b

bl

'06 '08 '10 '12 '14 '160

0.5

1

1.5

2x 10

9

years

NP

V, $

Economic limit: well water cut < 95% Implementation: lumped nonlinear constraint in optimization

Sept 18

10 20 30 40 50 60 70 801

2x 10

9

NP

V

objective evaluations

MMA

10 20 30 40 50 60 70 80

0.9550.96

0.97

wat

er c

ut

10 20 30 40 50 60 70 80 90 100 110

1.21.31.41.51.61.7

x 109

NP

V

objective evaluations

SQP

0.9550.96

0.97

wat

er c

ut

NTNU 2014

HM and PCA-based parameterization


NTNU 2014

Model: 28 × 30, 3-phase black-oil [Shirangi, 2011] Synthetic history: phase rates 10 × 10 days 400 permeability realizations used to provide PCA of

transmissibility 1,622 cell transmissibilities to reconstruct Initial guess: const (mean over field)

History matching: 2D Example

5 10 15 20 25

5

10

15

20

25

30

0

1

2

3

4

5

6

5 10 15 20 25

5

10

15

20

25

0

1

2

3

4

5

6


NTNU 2014

PCA transform uses 78 components (95% of total variance)

Main geological features captured

PCA-based Parameterization

5 10 15 20 25

5

10

15

20

25

30

0

1

2

3

4

5

6

5 10 15 20 25

5

10

15

20

25

0

1

2

3

4

5

6

5 10 15 20 25

5

10

15

20

25

30

0

1

2

3

4

5

6

5 10 15 20 25

5

10

15

20

25

0

1

2

3

4

5

6

0 100 200 300 400 500102

104

106

108

gradient evaluations

obje

ctiv

e fu

nctio

n


NTNU 2014

HM of geological features


NTNU 2014

Challenges

Large size of TRAN maps for 3D models

Existence of geological features (faults, barriers, etc.) ⇒ complicates the use of specific models and correspondent techniques, e.g. Gaussian

Lack of available realizations

Uncertainty in location and/or properties of geological features ⇒ available realizations in general may not be geologically consistent

History matching of real fields


NTNU 2014

Grouping transmissibilities given fault geometry 91 controls ⇒ 8 new parameters (5 faults + 3 intersections) as

multipliers to fault transmissibilities (ranging from 0.0001 to 10) History window shrunk to 10 days (16 data points out of 165) Initial guess is 1.0 (no fault)

Geology Properties: Faults

5 10 15 20 25

5

10

15

20

25

30

0

1

2

3

4

5

6

fault geometry

0.1

0.01

0.01

0.01

10

0.001

0.0010.0001

0 5 10 15 20 250

5

10

15

20

25

30


NTNU 2014

Faults: Reconstruction

Global minimum found short HM window no water breakthroughs “bad” initial guess

5 10 15 20 25

5

10

15

20

25

30

0

1

2

3

4

5

6

5 10 15 20 25

5

10

15

20

25

0

1

2

3

4

5

6

0 10 20 30 40 50 60 70 8010

-20

10-15

10-10

10-5

100

105


obje

ctiv

e fu

nctio

n


NTNU 2014

Model: 16 layers from 5th to 20th History: synthetic based on real

well schedule at the beginning of 2007 (phase rates)

Norne Field: faults

faults well perforation (current layer) well perforation (other layers) active cells

Layer #5

0 5 10 15 20 25 30 35 40 45 500

20

40

60

80

100


NTNU 2014

Example: barrier transmissibility (multiplier) history matching

8 barriers chosen manually 1,825 controls ⇒ 8 new

parameters, multipliers to barrier transmissibilities (ranging from 0.00003 to 0.25)

Initial guess: constant value 0.001

History: synthetic based on real well schedule during January 2007 (36 data points, phase rates)

Norne field: Barriers Layer #8

0 5 10 15 20 25 30 35 40 45 500

20

40

60

80

100


NTNU 2014

Norne field: Barriers Layer #10

0 5 10 15 20 25 30 35 40 45 500

20

40

60

80

100

Layer #15

0 5 10 15 20 25 30 35 40 45 500

20

40

60

80

100


NTNU 2014

Barrier results Barrier Name Size μ (real) μ (opt) error, % (opt)

Layer_8 168 0.02 0.02003 + 0.14 MZ_layer_10_C-segm_mid 60 0.05 0.04919 – 1.63 MZ_layer_10_C-segm_middle 588 0.25 0.24954 – 0.19 MZ_layer_10_E1 125 0.05 0.05005 + 0.09 MZ_layer_15_C_south 264 0.00003 0.0000301 + 0.46 MZ_layer_15_C_middle 421 0.00005 0.0000497 – 0.57 MZ_layer_15_E-1H 54 0.005 0.00482 – 3.62 MZ_layer_15_D-segm 145 0.01 0.01001 + 0.12

1 825 max | err | = 3.62 %

0 10 20 30

100

102

104

106


obje

ctiv

e fu

nctio

n

0 5 10 15 20 250

0.5

1

1.5

2

2.5

3

3.5x 104

accu

mul

ated

gra

dien

t

all barriers

all barriersselected barriers


NTNU 2014

HM with multiple data types


NTNU 2014

HM with multiple data types


HM objective function for different types of data

production phase rates

well bottom-hole pressure

phase saturation (proxy from seismic

observations)

Production (dynamic) data: limited amount + local sensitivity Seismic data: large size + phase -dependent sensitivity How to scale , and to enhance efficiency?

NTNU 2014

Multiobjective HM


How to scale objective coefficients to enhance efficiency?

Normalizing by phase relative density (changing volumetric rates to mass rates)

Equalizing initial contributions to the objective

Analysis by Pareto efficiency (optimality)

Objective scalarization by Multicriteria Decision Making Scheme (undergoing research)

NTNU 2014

Production vs. Seismic data


28 × 30 black-oil 3-phase model [Shirangi, 2011]

Initial saturation Data: production 230 pts,

seismic 3,360 pts 127 PCA components

(build on 400 realizations)

5 10 15 20 25

5

10

15

20

25

30

0

1

2

3

4

5

6

7

8

9

10

5 10 15 20 25

5

10

15

20

25

30

0

1

2

3

4

5

6

7

8

9

10

5 10 15 20 25

5

10

15

20

25

30

0

1

2

3

4

5

6

7

8

9

10

5 10 15 20 25

5

10

15

20

25

30

0

1

2

3

4

5

6

7

8

9

10

0 50 100 150 20010-1

100

101

102

103

iterations

production dataseismic dataprod & seismic

NTNU 2014

Pareto optimality


Convex hull built up only if convergence to Pareto front is guaranteed

Property of each type of data should be taken into account

Gradient-based optimization requires efficient multiple data assimilation techniques based on data structure analysis

production part,

seis

mic

par

t,

Pareto front points 10

410

5

100

101

NTNU 2014

Closed-loop reservoir management


NTNU 2014

Reservoir management


Production System

Data



Optimization

Controls

Optimization

Gradient Framework

Geological Model

Controls

Production system

Production Data

Seismic Data

• Reducing the uncertainty of the reservoir model and taking decision regarding long-term or short-term reservoir management

NTNU 2014

Example: Brugge field


synthetic reservoir by TNO [Peters et. al., 2010] 44,550 active gridblocks (139 × 48 × 9) 10 injectors & 20 producers (smartwells) 104 geological realizations available

NTNU 2014

Reservoir management


Reservoir 6 layers from 3rd to 8th, 2-phase dead-oil model Wells BHP controlled 10 injectors (water) & 20 producers True model realization #73

Target optimal production schedule based on updated reservoir

Production Optimization History Matching Production cycle: ~ 10 years (3,420 days)

549 BHPs (every 180 days)

Objective: NPV – oil 75 $/bbl, water 1$/bbl (inj/prod), discount 10%

History: phase rates & saturation based on and current

Initial guess : realization #101

82,582 TRAN multipliers

95 PCA components built on 104 mesh-consistent TRAN realizations

NTNU 2014

CLRM: Convergence


2 4 6 8 100.85

0.9

0.95

1

1.05

1.1

1.15x 10

10

closed loop iteration, n

NP

V, $

4 6 8 101.148

1.15

1.152x 10

10

Optimizer: SNOPT (PO & HM) Termination: max 20 gradient evaluations + optimality conditions Performance: @ 5th year

NTNU 2014

PO: Robustness


4 5 6 7 8 9 101.02

1.04

1.06

1.08

1.1

1.12

1.14

1.16x 10

10

years

NP

V, $

0 2 4 6 8 100

2

4

6

8

10

12x 10

9

years

NP

V, $

Notation: optimal (reference) schedule assuming known

optimal schedules using different initial guesses

NTNU 2014

Sweep efficiency: layer 5


25 30 35 40 4530

40

50

60

70

80

90

100

110

25 30 35 40 4530

40

50

60

70

80

90

100

110

25 30 35 40 4530

40

50

60

70

80

90

100

110

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

NTNU 2014

Questions? Thank you


smart fields consortium: optimization of oil field ... fields consortium: optimization of oil field...

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