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Ruslan Miftakhov

Master of Science in Petroleum Engineering

Joined FURSST group in Aug 2012

ℛ 𝒰𝑛+1; 𝒰𝑛, Δ𝑡 = 0

Chose Δ𝑡 Perform Newton iterations

If failed, chop Δ𝑡 and try again!

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𝝏𝑺

𝝏𝒕+𝝏𝑭

𝝏𝒙= 𝟎

v

𝓡 = 𝑺𝒏+𝟏 − 𝑺𝒏 +∆𝒕

𝑽𝑭(𝑺𝒏+𝟏) − 𝑭𝒊𝒏𝒋

𝑆 ∆𝑡𝑓𝑖𝑛𝑎𝑙

| ℛ |

0

𝑞𝑖𝑛𝑗 𝐹(𝑆)

𝑆

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Failed

𝝏𝑺

𝝏𝒕+𝝏𝑭

𝝏𝒙= 𝟎

v

𝓡 = 𝑺𝒏+𝟏 − 𝑺𝒏 +∆𝒕

𝑽𝑭(𝑺𝒏+𝟏) − 𝑭𝒊𝒏𝒋

𝑆 ∆𝑡𝑓𝑖𝑛𝑎𝑙

| ℛ |

0

𝑞𝑖𝑛𝑗 𝐹(𝑆)

𝑆

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Converged

𝑆 ∆𝑡𝑓𝑖𝑛𝑎𝑙

| ℛ |

0

Δ𝑡

v

v

𝝏𝑺

𝝏𝒕+𝝏𝑭

𝝏𝒙= 𝟎

𝓡 = 𝑺𝒏+𝟏 − 𝑺𝒏 +∆𝒕

𝑽𝑭(𝑺𝒏+𝟏) − 𝑭𝒊𝒏𝒋

𝑞𝑖𝑛𝑗 𝐹(𝑆)

𝑆

Safeguards

Classical

Trust Region

Line Search

Heuristic

Eclipse Appleyard (EA)

Modified Appleyard (MA)

Modified Trust-region (X. Wang)

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Stiff Physics:

Thermal Simulation

Compositional Simulation

Coupled with geomechanics

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𝒰 Δ𝑡

ℛ

v

v

v

vv

𝜆

Zero level curve: ℛ 𝒰 𝜆 ,Δ𝑡 𝜆 = 0

Parameterized curve: 𝑑ℛ

𝑑𝜆= 𝒥

d𝒰

𝑑𝜆+𝜕ℛ 𝒰𝑛+1,𝒰𝑛;Δ𝑡

𝜕Δ𝑡

dΔ𝑡

𝑑𝜆= 0

𝑡 =𝑑𝒰

𝑑𝜆= −𝒥−1

𝜕ℛ

𝜕Δ𝑡

𝑑Δ𝑡

𝑑𝜆

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Drawbacks:

1. Convergence neighborhood

2. Too many residual evaluations

3. No optimal steplength, 𝛼

Choice of parameterization

Order of approximation accuracy

Robust steplength selection

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Δ𝑡 parameterization:

-Requires no additional equation

Parameterization

unknown

Residual Equation Parameterization

𝜆 ℛ 𝒰 𝜆 , Δ𝑡 𝜆 = 0 𝑑ℛ

𝑑𝜆=𝜕ℛ

𝜕𝒰

d𝒰

𝑑𝜆+𝜕ℛ

𝜕Δ𝑡

dΔ𝑡

𝑑𝜆= 0

Δ𝑡 ℛ 𝒰 Δ𝑡 , Δ𝑡 = 0 𝑑ℛ

dΔ𝑡=𝜕ℛ

𝜕𝒰

d𝒰

𝜕Δ𝑡+𝜕ℛ

𝜕Δ𝑡= 0

𝒰

Δ𝑡

ℛ

𝜆12

Choice of parameterization

Order of approximation accuracy

Robust steplength selection

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Taylor series expansion :

𝒰pred = 𝒰0 + Δ𝑡𝑑𝒰

𝑑Δ𝑡+Δ𝑡2

2!

𝑑2𝒰

𝑑Δ𝑡2+Δ𝑡3

3!

𝑑3𝒰

𝑑Δ𝑡3+Δ𝑡4

4!

𝑑4𝒰

𝑑Δ𝑡4…

Order of

approximation

Terms

(1) Zero Order 𝒰pred = 𝒰0

(2) First Order𝒰pred = 𝒰0 + Δ𝑡

𝑑𝒰

𝑑Δ𝑡

(3) Second Order𝒰pred = 𝒰0 + Δ𝑡

𝑑𝒰

𝑑Δ𝑡+Δ𝑡2

2!

𝑑2𝒰

𝑑Δ𝑡2

Δ𝑡

(1) (2) (3)

𝒰 1 𝒰 2 𝒰 3

Solution

path

𝒰𝑛+1

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Parameterization

unknown

First order

approximationSecond order approximation

Δ𝑡 𝑑𝒰

𝑑Δ𝑡= −𝒥−1

𝜕ℛ

𝜕Δ𝑡

𝑑2𝒰

𝑑Δ𝑡2= −𝒥−1

𝜕𝒥

𝜕𝒰⨂𝑑𝒰

𝑑Δ𝑡+𝜕𝒥

𝜕Δ𝑡

𝑑𝒰

𝑑Δ𝑡+𝜕𝐺

𝜕𝒰

𝑑𝒰

𝑑Δ𝑡+𝜕𝐺

𝜕Δ𝑡

𝐺 =𝜕ℛ

𝜕Δ𝑡⨂ - Tensor-vector multiplication

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Δ𝑡 = 0.8

(1) (2) (3)

𝒰 1 𝒰 2 𝒰 3

Solution

path

𝒰𝑛+1

Order of

approximationTerms

Predicted

valueError, 𝜺𝟎

(1) Zero Order Spred = 𝑆0 0.0 -0.6255

(2) First Order 𝑆pred = 𝑆0 + Δ𝑡𝑑𝑆

𝑑Δ𝑡0.80 0.1745

(3) Second Order 𝑆pred = 𝑆0 + Δ𝑡𝑑𝑆

𝑑Δ𝑡+Δ𝑡2

2!

𝑑2𝑆

𝑑Δ𝑡20.7488 0.1233

𝑞𝑖𝑛𝑗 𝐹(𝑆)

𝑆𝑖𝑛𝑖𝑡 = 0.0

𝑆𝑜𝑙𝑢𝑡𝑖𝑜𝑛 𝑆𝑛+1 = 0.6255

𝜀0

Choice of parameterization

Order of approximation accuracy

Robust steplength selection

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𝛼

𝛼

𝒰

Δ𝑡

ℰ0( 𝛼)

𝑡1

𝑡21 2 3 4 5

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𝛼

𝛼

𝒰

Δ𝑡

𝑡1

𝑡21 2 3 4 5

𝜀

ℰ0( 𝛼)

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𝛼

𝛼

𝒰

Δ𝑡

ℰ0( 𝛼)

𝑡1

𝑡2

1 2 3 4

𝜀

1 2 3 4 5

𝜀

𝑈 𝑡 + Δ𝑡 = 𝑈 𝑡 + 𝛼𝑑𝑈 𝑡

𝑑Δ𝑡+𝛼2

2!𝑘 𝑡 𝑁 𝑡 + 𝑂 𝛼3

𝑈𝑝𝑟𝑒𝑑 = 𝑈 𝑡 + 𝛼𝑑𝑈 𝑡

𝑑Δ𝑡

ℰ0 = 𝑈𝑘 − 𝑈𝑝𝑟𝑒𝑑 ≈𝛼2

2!𝑘 𝑡 𝑁 𝑡

Take 𝛼2

2!𝑘 𝑡 𝑁 𝑡 ≤ 𝛼

𝑑𝑈 𝑡

𝑑Δ𝑡

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𝛼

𝑈𝑘 ≈ 𝑈∞ 𝑈0

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𝜑 𝜀 =𝜀 + 10 − 𝜀2

5 − 𝜀2

𝜑 𝜀 =𝜀2

3 − 2𝜀

𝜑 𝜀 = 𝜀𝑝

Assume Quadratic convergence

(Newton-Kantorovich):

SuperLinear error model:

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𝑄𝑢𝑎𝑑𝑟𝑎𝑡𝑖𝑐 𝑐𝑜𝑛𝑣𝑒𝑟𝑔𝑒𝑛𝑐𝑒 𝑚𝑜𝑑𝑒𝑙: 𝜑 𝜀 =𝜀 + 10 − 𝜀2

5 − 𝜀2

𝜀𝑖+1 ≤ 𝜑(𝜀𝑖)

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𝑆𝑢𝑝𝑒𝑟𝐿𝑖𝑛𝑒𝑎𝑟 𝑐𝑜𝑛𝑣𝑒𝑟𝑔𝑒𝑛𝑐𝑒 𝑚𝑜𝑑𝑒𝑙: 𝜑 𝜀 = 𝜀𝑝

𝜀𝑖+1 ≤ 𝜑(𝜀𝑖)

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1 2 3 4 5 6

Iterations

𝜀

Error

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1 2 3 4 5 6

Iterations

𝜀

Error

SubLinear

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1 2 3 4 5 6

Iterations

𝜀

Error

SubLinear

SuperLinear

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1 2 3 4 5 6

Iterations

𝜀

Error

SubLinear

SuperLinear

Quadratic

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𝜀 𝑁 = 𝜀01 + exp 𝐵𝑀

1 + exp(−𝐵(𝑁 −𝑀)), 𝑤ℎ𝑒𝑟𝑒 𝐵 𝑎𝑛𝑑 𝑀 𝑎𝑟𝑒 𝑓𝑖𝑡𝑡𝑖𝑛𝑔 𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡𝑠

𝐵 𝑑𝑒𝑝𝑒𝑛𝑑𝑎𝑛𝑐𝑦 𝑀 𝑑𝑒𝑝𝑒𝑛𝑑𝑎𝑛𝑐𝑦

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𝜀 𝑁 = 𝜀01 + exp 𝐵𝑀

1 + exp(−𝐵(𝑁 −𝑀))

𝜀𝑖+1 ≤ 𝜀𝑖 β(𝐾)

Parameterization parameters

Order of Approximation accuracy

Robust Steplength selection

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𝑡1

𝑡2

𝑡3Δ𝑡𝑡𝑎𝑟𝑔𝑒𝑡

Δt = 0

𝒰n+1𝑆0 = (𝒰n, 0)

𝑆2

𝑆3

Solution path

𝑆1Tiny initial

steplength

Steplength from

Adaptation

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1D Buckley Leverett simulation :

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𝐻𝑒𝑡𝑒𝑟𝑜𝑔𝑒𝑛𝑒𝑜𝑢𝑠 𝑔𝑟𝑖𝑑 𝑤𝑖𝑡ℎ𝐷𝑖𝑚𝑒𝑛𝑠𝑖𝑜𝑛 𝑜𝑓 60 × 220 × 50

6 𝑃𝑟𝑜𝑑𝑢𝑐𝑡𝑖𝑜𝑛 𝑊𝑒𝑙𝑙𝑠𝑃𝑤𝑓 = 1500 𝑝𝑠𝑖.

2 𝐼𝑛𝑗𝑒𝑐𝑡𝑖𝑜𝑛 𝑊𝑒𝑙𝑙𝑠𝑃𝑖𝑛𝑗 = 5000 𝑝𝑠𝑖.

𝐼𝑛𝑐𝑙𝑢𝑑𝑒𝑑 𝐺𝑟𝑎𝑣𝑖𝑡𝑦 𝑎𝑛𝑑𝐶𝑎𝑝𝑖𝑙𝑙𝑎𝑟𝑦 𝑒𝑓𝑓𝑒𝑐𝑡𝑠

P1

P2

P3

P4

P5

P6

Inj1

Inj1

http://www.spe.org/

P1

P2

P3

P4

P5

P6

Inj1

Inj1

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Modified Continuation-Newton:

Never cuts the timestep due to lack of convergence;

Devised a robust and automatic steplength selection strategy;

Showed that high-order CN methods are not competitive;

Future work:

Test on complex physics;

Implement in GENSOL.

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Ruslan Miftakhov

Master of Science in Petroleum Engineering

Joined FURSST group in Aug 2012