Control and optimzation of sub-surface flow

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Control and optimization of

subsurface flow

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SIAM CSE 2013

Subsurface flow

• Fluids

• Clean groundwater – drinking, irrigation

• Polluted ground water – clean-up, remediation • Oil – many components, viscous tar or water-like • Natural gas – may contain pollutants (e.g. H₂S) • CO₂ – liquid, gaseous, supercritical; dissolution

• Porous medium

• Porous soil (shallow) – sand, clay

• Porous rock (deep) – clastics (sandstones, shales), carbonates • Heterogeneous – layers, fractures, faults, dissolved, cemented • Interfacial processes – surface tensions, adsorption, desorption

• Environment

• High pressures: 10 MPa / km

• High temperatures: 25 o_{C / km} 3000 m: 300 bar, 100

o_C

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fluids trapped in porous rock below impermeable ‘cap rock’

Oil & gas reservoirs

gas oil

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Oil production mechanisms

• Primary recovery – expansion of rock and fluids,

decreasing reservoir pressure

depletion drive, compaction drive, 5-40% recovery

• Secondary recovery – injection of fluids to maintain

reservoir pressure and displace oil to the wells

water flooding, gas flooding, 10-60% recovery

• Tertiary recovery – injection of heat or chemicals to

change physical properties (e.g. viscosity, wettability)

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Research & development drivers

• Increasing demand; reducing supply

• energy demand continues to grow world-wide

• renewables are developing too slow to keep up with demand • ‘easy oil’ has been found; few new discoveries; complex fields

=> produce more from existing reservoirs

• Increasing knowledge- and data intensity

• more sensors: pressure/temperature/flow, time-lapse

seismics, passive seismics, EM, tilt meters, remote sensing, … • more control: multi-lateral wells, smart wells,

snake wells, dragon wells, remotely controlled chokes, … • more modeling capacity: computing power, visualization

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Governing equations – simple example

• Oil and water only, no gravity, no capillary pressures

• Separate equations for

p and

S

_w

:

•

l

_t,

_c

_t and

_f

_w are functions of

_S

;

_v

_t is a function of

_p

• Coupled and nonlinear, (near-)elliptic, (near-)hyperbolic

 

w t w w w

S

v f

S

q

t













2 t t t

p

c

q

t

l







 





K

diffusion

convection

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Reservoir simulation

• 3-phases (gas, oil, water) or multiple components

+ thermal effects + chemical effects + geo-mechanics + …

• Nonlinear PDEs discretized in time and space – FD/FV

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Reservoir simulation

• 3-phases (gas, oil, water) or multiple components

+ thermal effects + chemical effects + geomechanics + …

• Nonlinear PDEs discretized in time and space – FD/FV

• Cornerpoint grids or unstructured grids

• Large variation in parameter values: 10-15_<

_k

_{< 10}-11_m2

• Typical model size: 104_–106_{cells, 50–500 time steps}

• Fully implicit (Newton iterations) – clock times: hours-days

• Typical code size: 106_-107_{lines (well models, PVT analysis)}

• Research focused on upscaling, gridding, ‘history matching’

(inverse modeling), new physics, solvers, parallelization

• Primarily used in design phase: field (re-)development

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Closed-loop reservoir management

• Hypothesis: recovery can be significantly increased by

changing reservoir management from a ‘batch-type’ to a near-continuous model-based controlled activity

• Key elements:

• Optimization under geological uncertainties

• Data assimilation for frequent updating of system models

• Inspiration:

• Systems and control theory

• Meteorology and oceanography

• A.k.a. real-time reservoir management, smart fields,

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Closed-loop reservoir management

Noise Input System Output (reservoir, wells

& facilities)

Noise

Sensors

Predicted output Measured output Data

assimilation algorithms Controllable

input

System models well logs, well tests, Geology, seismics, fluid properties, etc.

Optimization algorithms

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Open-loop flooding optimization

Data assimilation

algorithms

Noise Input System Output Noise (reservoir, wells & facilities) Optimization algorithms Sensors System model

Predicted output Measured output

Controllable input

Geology, seismics, well logs, well tests, fluid properties, etc.

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Optimization techniques

• Global versus local

• Gradient-based versus gradient-free

• Constrained versus non-constrained

• ‘Classical’ versus ‘non-classical’ (genetic algorithms,

simulated annealing, particle swarms, etc.)

• We use ‘optimal control theory’

• Gradient-based – steepest ascent, LBFGS, trust regions, …

• Gradients obtained with ‘adjoint’ equation (implicit differentiation) • Computational effort independent of number of controls

• Objective function: ultimate recovery or monetary value

• Controls: injection/production rates, pressures or valve openings (102 _{to 10}3_{control variables, 10}4_{– 10}6_{states )}

• Constraint handling: GRG, lumping, SQP, augmented Lagrangian, … • Beautiful, but code-intrusive and requires lots of programming

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12-well example

• 3D reservoir

• High-permeability channels

• 8 injectors, rate-controlled

• 4 producers, BHP-controlled

• Production period of 10 years

• 12 wells x 10 x 12 time steps

gives 1440 optimization parameters

u

• Optimization of monetary value

J

• Adjoint gives us:

Van Essen et al., 2006

J

= (value of oil – costs of water produced/injected)

1 2 1440

dJ

J

d

u



_



 

_







u

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Why this wouldn’t work

• Real wells are sparse and far apart

• Real wells have more complicated constraints

• Field management is usually production-focused

• Long-term optimization may jeopardize short-term profit

• Optimal inputs cannot be implemented (too dynamic)

• Production engineers don’t trust reservoir models anyway

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Robust open-loop flooding optimization

Data assimilation

algorithms

Noise Input System Output Noise (reservoir, wells & facilities) Optimization algorithms Sensors System models

Controllable input

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Robust optimization

• Use ensemble of realizations (typically 100)

• Optimize expected value over ensemble

• Single strategy, not 100!

• If necessary include risk aversion (utility function)

• Computationally intensive

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SIAM CSE 2013

Robust optimization results

3 control strategies applied to set of 100 realizations:

reactive control, nominal optimization, robust optimization

Fig. 3: Cumulative Distribution Function and Probability Density Functions based on the first set of 100 realizations of the reactive control strategy, the 100 nominal optimization strategies and the

Monetary value (M$) Pr obabil ty dens ity

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Closed-loop flooding optimization

& facilities)

Noise

Sensors

input

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Data assimilation

(‘computer-assisted history matching’)

• Uncertain parameters, not initial conditions or states

• Parameters: permeabilities, porosities, fault multipliers, …

• Data: production (oil, water, pressure), 4D seismics, …

• Very ill-posed problem: many parameters, little info

• Variational methods – Bayesian framework:

• Ensemble Kalman filtering – sequential methods

• Reservoir-specific methods (e.g. streamlines)

• ‘Non-classical’ methods – simulated annealing, GAs, …

• Monte Carlo methods – MCMC with proxies





1



 



1





0 0

T T

d m

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Closed-loop flooding optimization

‘Truth model’

& facilities)

Noise

Sensors

input

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System models well logs, well tests, Geology, seismics, fluid properties, etc. ‘Truth model’

Noise

Sensors Noise Input System Output

(reservoir, wells & facilities) Controllable input Optimization algorithms etc., etc.

assimilation algorithms

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Closed-loop

–

effect of cycle time

1 2 3 4 5 6 8.5 9 9.5 10 10.5x 10 7 N PV , $ 1 2 3 4 5 6 -2 -1.5 -1.0 -0.5 0 D is cou nt ed w at er cos ts , $ 1 2 3 4 5 6 8.5 9 9.5 10 10.5x 10 7 D is cou nt ed oil rev en ue s, $ reactive open-loop

1 month 1 year 2 years 4 years

Hypothesis: recovery can be significantly increased by

changing reservoir management from a ‘batch-type’ to a near-continuous model-based controlled activity

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Link with short-term optimization

Data assimilation

algorithms

Controllable input

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• Life-cycle optimization attractive for reservoir engineers

• Not so attractive for production engineers:

• Decreased short term production

• Erratic behavior of optimal operational strategy

+8%

Net Present Value - No Discounting

time [year] R eve n u es [ M $ ] Reactive Control Optimal Control -50%

short term horizon

Reactive control & optimal control

injector 1 time [year] flow rate [bb l/d] injector 2 time [year] flow rate [bb l/d] injector 3 time [year] flow rate [bb l/d] injector 4 time [year] flow rate [bb l/d] injector 5 time [year] flow rate [bb l/d] injector 6 time [year] flow rate [bb l/d] injector 7 time [year] flow rate [bb l/d] injector 8 time [year] flow rate [bb l/d] producer 1 time [year] flow rate [bb l/d] producer 2 time [year] flow rate [bb l/d] producer 3 time [year] flow rate [bb l/d] producer 4 time [year] flow rate [bb l/d]

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Hierarchical optimization

• Optimize objectives sequentially

• Optimal first objective constrains second optimization

• Only possible if there are redundant degrees of freedom in

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Hierarchical optimization

• Order objectives in relative importance

• Optimize objectives sequentially

• Optimal first objective constrains second optimization

• Only possible if there are redundant degrees of freedom in

input parameters after meeting primary objective

• Second optimization is performed in null space of control

variables (directions in which

_dJ/du

remains constant)

• Null space defined by Hessian

d

2

J/du

2

• Small deviations from optimal first objective should be

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Hierarchical optimization: example

•Same model as before

•Rates in 8 injectors optimized

•Primary objective:

life-cycle monetary value

•Secondary objective:

strongly discounted monetary value (25%) to emphasize

short term production value

•Hessian approximated by adjoint (first-order derivatives)

in combination with finite differences

•Alternatives: approximate Hessian with LBFGS or switching

between two objectives

Van Essen et al., 2006

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Hierarchical optimization: results

50 100 150 200 24 25 26 27 28 29 30 31 32 Iterations N et P re se n t V a lu e - D isc o u n te d [M $ ]

Secondary Objective Function

50 100 150 200 40 41 42 43 44 45 46 47 48 Iterations N et P re se n t V a lu e - U n d isc o u n te d [M $ ]

Primary Objective Function

Optimization of secondary objective function - constrained to null-space

of primary objective 20 40 60 80 100 24 25 26 27 28 29 30 31 32 Iterations N et P re se n t V a lu e - D isc o u n te d [M $ ]

Secondary Objective Function

20 40 60 80 100 40 41 42 43 44 45 46 47 48 Iterations N et P re se n t V a lu e - U n d isc o u n te d [M $ ]

Primary Objective Function

Optimization of secondary objective function - unconstrained +28.2% -0.3% +28.2% -5.0%

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Hierarchical optimization: results

0 900 1800 2700 3600 0 5 10 15 20 25 30 35 40 45 50 time [days] NP V ov er T im e - U ndi sc ount ed [10 6 $] ~ ~

value of objective function J

1 resulting from u *_. value of objective function J

1 resulting from u * value of objective function J

1 resulting from u Time (d) Monetary value (M$) 0 3600 0 30 900 1800 2700 15

life cycle optimization

unconstrained optimization

hierarchical optimization

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Subsurface flow controllability

Data assimilation

algorithms

Controllable input

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System-theoretical concepts

• Controllability of a dynamic system is the ability to

influence the states through manipulation of the inputs.

• Observability of a dynamic system is the ability to

determine the states through observation of the outputs.

• Identifiability of a dynamic system is the ability to

determine the parameters from the input-output behavior.

• Well-defined theory for linear systems. More difficult for

nonlinear ones. System model state (p,S) parameters (k,,…) output (p_wf,q_w,q_o) input (p_wf,q_t)

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System-theoretical concepts

• Controllability of a dynamic system is the ability to

influence the states through manipulation of the inputs.

• Observability of a dynamic system is the ability to

determine the states through observation of the outputs.

• Identifiability of a dynamic system is the ability to

determine the parameters from the input-output behavior.

• Want to know more?

Attend MS14 Today, 9:30 AM - 11:30 AM Grand Ballroom A

System model state (p,S) parameters (k,,…) output (p_wf,q_w,q_o) input (p_wf,q_t)

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SIAM CSE 2013

System theory – main findings so far

• Controllablity, observability and identifiability are very

limited

• Reservoir dynamics ‘lives’ in a state space of a much

smaller dimension than the number of model grid blocks

• Linear case (pressures only): typical number of relevant

pressure states: 2 x # of wells

• For fixed wells: the (few) identifiable parameter patterns

correspond just to the (few) controllable state patterns

• Scope for reduced-order modeling to speed up iterative

optimization, history matching, upscaling?

• First attempts: POD – disappointing speed-ups • Successful: TPWL (Durlofsky et al.)

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System theory – main findings so far

• Controllablity, observability and identifiability are very

limited

• Reservoir dynamics ‘lives’ in a state space of a much

smaller dimension than the number of model grid blocks

• Linear case (pressures only): typical number of relevant

pressure states: 2 x # of wells

• For fixed wells: the (few) identifiable parameter patterns

correspond just to the (few) controllable state patterns

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System theory – main findings so far

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System theory – main findings so far

• Interpreting the ‘history matched’ results requires

geological insight

• Understanding optimization results also requires

geological insight

• Well location-optimization requires a geological model

• However, we need to focus on the relevant geology:

 Which geological features are identifiable?

 Which geological features influence controllability?

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Conclusions, questions, more work

• Size of the prize still unclear (open- and closed-loop)

• Adjoint based-optimization techniques work well;

constraints, regularization, efficiency still to be improved

• Specific optimization methods less important than

workflow & human interpretation of results

• Field acceptance of closed-loop approach will require

combination with short-term production optimization

• Reservoir dynamics lives in low-order space. Wide scope

for reduced-order, control-relevant modeling

• Control-relevant geology – how do we define it?

• Current focus:

• Expansion to EOR (thermal, compositional, chemical flooding)

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Acknowledgments

• Collaborators:

Okko Bosgra†

Arnold Heemink Paul Van den Hof

and many other colleagues and students of

• TU Delft – Department of Geoscience and Engineering • TU Delft – Delft Center for Systems and Control

• TU Delft – Delft Institute for Applied Mathematics • TU Eindhoven – Department of Electrical Engineering • TNO – Built Environment and Geosciences

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