A new approach for dynamic optimization of water flooding problems

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A new approach for dynamic

optimization of water flooding problems

Rolf J. Lorentzen

Aina M. Berg Geir Nævdal Erlend H. Vefring

IRIS – International Research Institute of Stavanger (formerly Rogaland Research)

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Overview

• Introduction

• Brief overview of the new methodology

• Brief overview of the Partial Enumeration Method (PEM)

• Example – New approach and PEM on a synthetic reservoir model

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Introduction

• Smart wells – remotely operated downhole chokes

• Controlling chokes – water flooding optimized • Maximizing cumulative oil production (COP)

or net present value (NPV)

• Avoids limitations – no adjoint equations needed

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Brief overview of the new

methodology

. 1 ], , , , [ ) ( , ] ) ( [m c c c₁ c₂ c i N U i ₌ i i T T i T ₌ i i K _Mi ₌ K

An set (ensemble) of (_N) state vectors is constructed

from choke settings and calculated COP or NPV

Here _mrepresents the total COP or NPV. The

production interval is divided into a set of _M

regulation intervals.

Choke settings are constant within each regulation interval.

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Brief overview of the new

methodology

The approach is motivated by the ensemble Kalman filter (EnKF), and is based on calculation of a Kalman gain matrix with zero measurement error. It can be

shown that in our case, this matrix is given by

, ) ˆ ( ) ˆ )( ˆ ( 1 1 2 1           − − − =

∑

= = N i i N i i i m m m m c c K

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Brief overview of the new

methodology

Each member (_i) of the ensemble is updated

according to the following formula

) ( ~ _, ₁_, ₁ _j ₁ _j ₁_,_i o j i j i j m D K U U ₌ − ₊ − − ₋ −

The difference between this approach and the

traditional Kalman filter update, is that the vector here represents an upper limit for the total COP or NPV. This value is calculated according to

1 − j o D ). ( std ) max( 1 1 1 ₋ ₋ − + = j j j o m m D

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1D interpretation

The figure shows how each member of the ensemble

is updated in the direction given by _K.

0.8 1 1.2 1.4 1.6 1.8 2 0 0.5 1 1.5 2 2.5 3 c m upper limit prior posterior upd. direction

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Updating procedure

• First data ( ) and choke settings ( ) are collected

• Run filter to produce and .

• Continuous choke settings ( ) are rounded to the closest allowed discrete setting to

produce .

• Forward simulations using are run to produce . 1 − j m c j−1 j m~ c~j j c~ j c j c j m

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Net present value

∑

= ∆ + ∆ − ∆ − ∆ = M k t k wi k wi wp k wp op k b p r p r p r J 1 ₃₆₀ 0 ) 100 / 1 ( b r r r p p p wi wp o wi k wp k op k ∆ ∆ ∆

The objective function for the NPV is given by the following formula

: Oil production during

: Water production during : Water injection during

: Benefit factor for oil production : Cost factor for water production : Cost factor for water injection : Interest rate (in percent)

t ∆ t ∆ t ∆

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Brief overview of the Partial

Enumeration Method

1. Iteration index k = 0. 2. Select choke j.

3. For choke j, do:

a. Switch to one of the allowed settings.

b. Run simulator for a given period of time.

c. Repeat a-b for all allowed settings and choose the setting that results in highest oil production.

4. Repeat 2-3 for all chokes. 5. Increase k by 1.

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Example

• Reservoir dimensions: 1020 m x 510 m horizontally and 50 m vertically.

• Reservoir divided into 30 x 3 x 20 grid blocks. • Five horizontal layers with thickness 10 m.

• Layers have permeability (mD) 100, 1000, 50, 750 and 50 from top to bottom.

• Vertical permeability between layers is 1% of horizontal.

• Two wells penetrating the reservoir, one producer and one injector.

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Example

• The producer has four inflow zones and the injector has five injection zones (which gives a total of nine chokes).

• Production chokes have three positions: open, half open and closed.

• Injection chokes have two positions: open and closed. • Maximum oil production is 2500 scm/day.

• Minimum bottom hole pressure for producer is 215 bar.

• Maximum bottom hole pressure for injector is 285 bar. • Water injection by voidage replacement (controlled by

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Filter variables

• Ensemble size is 100.

• Number of iterations is 31.

• Regulation interval is 180 days.

• Number of regulation intervals is 10 (which gives a total production interval of 5 years).

• Number of chokes is 9. Economic parameters

• Benefit factor for oil production: 50 $/bbl. • Cost factor for water production: 10 $/bbl. • Cost factor for water injection: 0 $/bbl.

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Development of optimized COP

Total COP vs. Iterations.

0 5 10 15 20 25 30 35 1.52 1.54 1.56 1.58 1.6 1.62 1.64 1.66 1.68 1.7 1.72x 10 6 Scm ref solution EnKF−COP

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Development of optimized NPV

NPV vs. iterations. 0 5 10 15 20 25 30 35 3.1 3.15 3.2 3.25 3.3 3.35 3.4 3.45 3.5 3.55 3.6x 10 8 $ EnKF−NPV

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Choke settings for EnKF-COP

500 1000 1500 Ch. 1 Days 500 1000 1500 Ch. 2 Days 500 1000 1500 Ch. 3 Days 500 1000 1500 Ch. 4 Days 500 1000 1500 Ch. 5 Days 500 1000 1500 Ch. 6 Days 500 1000 1500 Ch. 7 Days 500 1000 1500 Ch. 8 Days 500 1000 1500 Ch. 9 Days Closed Half open Open

Demo

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Choke settings for EnKF-NPV

Chokes 1-4: producer, 5-9: injector. 500 1000 1500 Ch. 1 Days 500 1000 1500 Ch. 2 Days 500 1000 1500 Ch. 3 Days 500 1000 1500 Ch. 4 Days 500 1000 1500 Ch. 5 Days 500 1000 1500 Ch. 6 Days 500 1000 1500 Ch. 7 Days 500 1000 1500 Ch. 8 Days 500 1000 1500 Ch. 9 Days Closed Half open Open

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Choke settings for PEM

Chokes 1-4: producer, 5-9: injector. 500 1000 1500 Ch. 1 Days 500 1000 1500 Ch. 2 Days 500 1000 1500 Ch. 3 Days 500 1000 1500 Ch. 4 Days 500 1000 1500 Ch. 5 Days 500 1000 1500 Ch. 6 Days 500 1000 1500 Ch. 7 Days 500 1000 1500 Ch. 8 Days 500 1000 1500 Ch. 9 Days Closed Half open Open

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Cumulative oil production

0 200 400 600 800 1000 1200 1400 1600 1800 0 2 4 6 8 10 12 14 16 18x 10 5 Days Scm EnKF−COP EnKF−NPV PEM ref solution

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Cumulative water production

0 200 400 600 800 1000 1200 1400 1600 1800 0 1 2 3 4 5 6 7x 10 6 Days Scm EnKF−COP EnKF−NPV PEM ref solution

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Cumulative water injection

0 200 400 600 800 1000 1200 1400 1600 1800 0 1 2 3 4 5 6 7 8 9x 10 6 Days Scm EnKF−COP EnKF−NPV PEM ref solution

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Water saturation after 1800 days

10 20 30 2 4 6 8 10 12 14 16 18 20 x z First layer 10 20 30 2 4 6 8 10 12 14 16 18 20 x z Second layer 10 20 30 2 4 6 8 10 12 14 16 18 20 x z Third layer

Left: EnKF-COP Right: PEM Oil: red, water: blue.

10 20 30 2 4 6 8 10 12 14 16 18 20 x z First layer 10 20 30 2 4 6 8 10 12 14 16 18 20 x z Second layer 10 20 30 2 4 6 8 10 12 14 16 18 20 x z Third layer

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Water saturation after 1800 days

Left: Reference Right: EnKF-NPV Oil: red, water: blue

10 20 30 2 4 6 8 10 12 14 16 18 20 x z First layer 10 20 30 2 4 6 8 10 12 14 16 18 20 x z Second layer 10 20 30 2 4 6 8 10 12 14 16 18 20 x z Third layer 10 20 30 2 4 6 8 10 12 14 16 18 20 x z First layer 10 20 30 2 4 6 8 10 12 14 16 18 20 x z Second layer 10 20 30 2 4 6 8 10 12 14 16 18 20 x z Third layer

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Comparison of the approaches

• PEM used 440 Eclipse simulations with duration 6

months.

• The EnKF used 3100 Eclipse simulations with duration 5 years.

• Number of forward simulations will increase rapidly for PEM when number of chokes and allowed

settings increase or when the regulation interval decrease.

• EnKF can easily be extended to handle variety of objective functions, and can be extended to handle continuous choke settings.

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Conclusions

• We have demonstrated a new approach for controlling downhole chokes so that water flooding is optimized.

• EnKF is used to maximize either total COP or NPV.

• Results are better compared to PEM.

• At the current stage, the EnKF is more time consuming.

• Further work: Faster convergence and extension to large scale field examples.