Reduced Order Model Applied to

Waterflooding Optimization

Manuel Fragoso Bernardo Horowitz

October 8th, 2013

• Linearize the model around previously stored states – approximation is valid in a confidence neighborhood.

• Physical surrogate model – replaces the simulation by a simple sequence of linear system solutions based on the physics of the problem.

• Jacobian and other derivative matrices are required in each stored snapshot.

Trajectory Piecewise Linearization - TPWL

Trajectory Piecewise Linearization - TPWL

Full-Implicit Discrete Flow Equation 1 1 1 1 1 1( , , ) ( , ) ( ) ( , ) 0n n n n n n n n+ + + + + += + + =g x x u A x x F x Q x u

Linearization around a stored state i: 1 1( , , )i i i+ +x x u1 1 1

1 1 1 1 1 11 1( ) ( ) ( ) 0

i i in i n i n i n i

i i i

+ + ++ + + + + +

+ +

∂ ∂ ∂= + − + − + − =

∂ ∂ ∂g g gg g x x x x u ux x u

Forecast solution – No iteration needed – Linear System Solution

( ) ( ) ( )1 1

1 1 1 1 11

i ii n i n i n i

i i

+ ++ + + + +

+

∂ ∂− = − − + − ∂ ∂

A QJ x x x x u ux u

11

1

ii

i

++

+

∂=∂gJx

where

TPWL reduces problem complexity. No dimension reduction yet! How to reduce problem dimension in order to further increase speedup?

• Takes advantage of the correlation between elements in a dataset – E. g.: Almost every person has on

their face a nose, a mouth, two eyes, two ears at similar positions.

• Despite of the high number of pixels in the image files, they are supposed to be represented in a quite reduced dimension space.

Proper Orthogonal Decomposition - POD

How POD works? What is it for?

Proper Orthogonal Decomposition - POD

a) A data set given as 3-dimensional points. b) The three orthogonal Principal Components (PCs) for the data, ordered by variance. c) The projection of the data set into the first two PCs, discarding the third one. (Kavraki, L.)

TPWL + POD

Applying POD – z is the projection of the state vector x into a lower dimension space

Where:

( ) ( )11 1

1 1 1 1 11( )

ii i

n i n i n ir i i

r r

++ +

+ + − + ++

∂ ∂= − − + − ∂ ∂

A Qz z J z z u ux u

≈xΦz

POD basis – derived from pressure and mass fraction maps as snapshots

1 1 1T T( )i i i

rJ J J+ + +

= Φ Φ

11 1

T T( )i

i i

i ir

A AJx x

++ + ∂ ∂

= Φ Φ ∂ ∂

11 1

T T1 1( )

ii i

i ir

Q QJu u

++ +

+ +

∂ ∂= Φ ∂ ∂

TPWL+POD

• Mass Fraction Formulation – Easier to extend to compositional – No problems with gas appearance/disappearance

• Two phase flow – oil & water

• Primary Variables:

Reservoir Simulator - SIMPAR

( ) ( ) ( ) ( ) ( ), ,

1 1 111 1, , ,

| 0 | 02 2l f l f

n n nn nil m l m l f f f l f f f l f fn i i i i i

f X f X

V z z X T X T X Mt

φρ φρ+ + ++

+ −≠ ≠

− = ∆Φ − ∆Φ + ∆ ∑ ∑

,o wp z

uku

kll

kk

xxxxxasxxFxf

≤≤≤≤

+=

..)()()(ˆmin δ

• Transforms optimization problem with the real model into a sequence of problems using surrogate model in a trust region to be solved by SQP algorithm

Multifidelity Sequential Approximate Optimization

min ( ). . l u

f xs a x x x≤ ≤ k k k

u ck k kl c

x xx x

= + ∆= −∆0,1,2,...k =

Real Model – High Fidelity

Phisical Surrogate Model – Low Fidelity Kriging Model

Trust Region

• Kriging Correction – Error correction based on real model responses – Latin Hypercube Sampling to choose the points

used to build the surface

( ) ( ) ( )

1

k

j jj

f N Zβ=

= +∑x x x

( ) ( ) ( )2cov , , ,i j i jZ Z σ = x x Rθ x x

Polinomial Error – Modeled by a Stochastic Process

LHS

Multifidelity Sequential Approximate Optimization

• SIMPAR – FORTRAN 77 & 90 – Developed by Petrobras Research Team (90’s)

– Source is available

• TPWL+POD & MSAO Algorithms – Matlab – Easier to prototype

– Suitable to Linear Algebraic Problems

• Kriging & LHS – DACE – IMM/DTU – Easy to use; Many Kriging Functionalities Implemented;

Matlab Toolbox

• Fortran-Matlab Link – HDF5: – Text Files: Extreme Slow; Massive Files (Abandoned)

– Hierarchical Data Format: Much Faster; Smaller Files – Available in many languages – Good Data Organization

Software Development

• 24000 grid blocks • Water & Oil

Densities are the same

• Training Simulation – 20 control Cycles – Prod. BHP: 4000 –

6000 psi – Inj. BHP: 9000 psi

TPWL Assesment

0 500 1000 1500 2000 2500 3000 3500 40004000

4200

4400

4600

4800

5000

5200

5400

5600

5800

6000

Test Case – Based on SPE10 Model

Kx

BHP Sequence

0 500 1000 1500 2000 2500 3000 3500 40000

500

1000

1500

2000

2500

3000

Oil Produced in Each Well

Simulator

TPWL

CPU Time Simpar – 2800s

TPWL – 7s Speed-up:400

NPV Simpar – $7.07x108 TPWL – $7.13x108

Error: 1.8%

Oil Produced Water Produced Water Injected Test Simulation – 2 Control Cycles

Very Good Surrogate Model to be used in a Multifidelity Optmization FrameWork.

TPWL Assesment

Compressibilities Rock – 10-7 psi-1 Water – 10-6 psi-1 Oil – 3x10-6 psi-1

NPV Simpar – $7.44x108 TPWL – $8.42x108

Error: 11.6%

Oil Produced Water Produced Water Injected Test Simulation – 2 Control Cycles

Kriging correction should be applied. TPWL may be re-trained if necessary.

TPWL Assesment

• Quarter five spot – 300 cells • 1 Producer and 1 Injector • Two-Phase Flow / Compressible fluid and Rock • Three layers with different permeabilities • Based on SPE1 Benchmark Case

• Injector BHP fixed – 10.000psi • Producer BHP controled – 4000-6000 psi • Oil Price - $100.00 / Inj & Prod Water Cost - $10.00 • 16000 days of simulation

Testing MSAO Algorithm

• NPV Optimization • X-Axis – Normalized BHP in the

first control cycle • Y-Axis – Normalized BHP in the

last control cycle • Theoretical optimum suposed

to be (0,1) • Confirmed using SQP over High

and Low Fidelity models • Confirmed by MSAO algorithm

Testing MSAO Algorithm

• Assesment of Optmization Schemes – TPWL re-training criteria – Continuous Refinement Scheme – Number of simulations to build kriging

correction

• Selection of test cases

– Quarter five-spot - SPE1 based (Very Small problem – suitable to test software)

– SPE10 based case – Already used in Opt Studies

– Brugge case – Under Construction

Current Work

• TPWL is suitable to sequential approximate opt algorithms due to its accuracy and speedup

• TPWL is supposed to be re-trained when large discrepancies occur.

• The multifidelity technique with continuous refinement of the correction (kriging) is expected to decrease the need for re-training.

• The use of HDF5 to exchange information is quite easy and efficient.

Conclusions

Thank you very much

mfragoso@petrobras.com.br mmfragoso@gmail.com