A new approach for dynamic optimization of water flooding problems

Transcription

1 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) 1

2 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 Summary

3 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 Model treated as a black box 3

4 Brief overview of the new methodology An set (ensemble) of (N) state vectors is constructed from choke settings and calculated COP or NPV U i i i T T i T i i i = [ m ( c ) ], ( c ) = [ c1, c, K, cm ], i = 1K N. Here m represents 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. We also have m i = f ( c i ), where f is the forward model (Eclipse in our case).

5 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 K = N i= 1 ( c N i i= 1 1 i cˆ)( m mˆ ), i ( m mˆ ) where ( ˆ) represents the mean value. 5

6 Brief overview of the new methodology Each member (i) of the ensemble is updated according to the following formula ~ j, i j 1, i j 1 j 1 j 1, i = U + K ( Do m 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 j 1 D o D = max( m j 1 j 1 j 1 o ) + std( m ).

7 1D interpretation 3.5 upper limit prior posterior upd. direction m c The figure shows how each member of the ensemble 7 is updated in the direction given by K.

8 Updating procedure j 1 j 1 First data ( m ) and choke settings ( c ) are collected j Run filter to produce m ~ j and c ~. j Continuous choke settings ( c ~ ) are rounded to the closest allowed discrete setting to j produce c. j Forward simulations using c are run to j produce m.

9 Net present value The objective function for the NPV is given by the following formula p p p ro rwp rwi b J op k wp k wi k = M op wp 0 k wp k wi t k = 1 k 30 r p (1 + r b p / 0 : Oil production during : Water production during : Water injection during ) r t t t : Benefit factor for oil production p : Cost factor for water production : Cost factor for water injection : Interest rate (in percent) wi k 9

10 Brief overview of the Partial Enumeration Method 1. Iteration index k = 0.. 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.. Repeat -3 for all chokes. 5. Increase k by 1.. Repeat -5 until convergence.

11 Synthetic reservoir model 11

12 Example Reservoir dimensions: 0 m x 5 m horizontally and 50 m vertically. Reservoir divided into 30 x 3 x 0 grid blocks. Five horizontal layers with thickness m. Layers have permeability (md) 0, 00, 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. 1

13 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 500 scm/day. Minimum bottom hole pressure for producer is 15 bar. Maximum bottom hole pressure for injector is 5 bar. Water injection by voidage replacement (controlled by reservoir fluid volumerate). 13

14 Filter variables Ensemble size is 0. Number of iterations is 31. Regulation interval is days. Number of regulation intervals is (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: $/bbl. Cost factor for water injection: 0 $/bbl. Interest rate: %. 1

15 Development of optimized COP 1.7 x Scm ref solution EnKF COP Total COP vs. Iterations. 15

16 Development of optimized NPV $ 3. x EnKF NPV NPV vs. iterations. 1

17 Choke settings for EnKF-COP Ch. 1 Ch. Ch. 3 Ch. Ch. Ch. 5 Ch. 7 Demo Ch. Closed Half open Open Ch. 9 Chokes 1-: producer, 5-9: injector. 17

18 Choke settings for EnKF-NPV Ch. 1 Ch. Ch. 3 Ch. Ch. Ch. 5 Ch. 7 Ch. Demo Closed Half open Open Ch. 9 Chokes 1-: producer, 5-9: injector. 1

19 Choke settings for PEM Ch. 1 Ch. Ch. 3 Ch. Ch. Ch. 5 Ch. 7 Ch. Demo Ch. 9 Closed Half open Open Chokes 1-: producer, 5-9: injector. 19

20 Cumulative oil production 1 x Scm EnKF COP EnKF NPV PEM ref solution

21 Cumulative water production 7 x EnKF COP EnKF NPV PEM ref solution 5 Scm

22 Cumulative water injection 9 x 7 EnKF COP EnKF NPV PEM ref solution Scm

23 Water saturation after 0 days First layer Second layer Third layer First layer Second layer Third layer z z z z z z x 0 30 x 0 30 x 0 30 x 0 30 x 0 30 x Left: EnKF-COP Right: PEM Oil: red, water: blue. 3

24 Water saturation after 0 days First layer Second layer Third layer First layer Second layer Third layer z z z z z z x 0 30 x 0 30 x 0 30 x 0 30 x 0 30 x Left: Reference Right: EnKF-NPV Oil: red, water: blue

25 Comparison of the approaches PEM used 0 Eclipse simulations with duration months. The EnKF used 30 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. 5

26 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.