Efficient Method of Solution of Large Scale

Engineering Problems with Interval Parameters Based

on Sensitivity Analysis

Andrzej PownukSilesian University

of Technology, Poland

2/79

Slightly compressible flow- 2D case

tp

B

cVqy

yp

B

kA

yx

xp

BkA

x oc

bsc

yycxxc

)(1 o

o

ppc

BB

),,(),( * txptxpp

),,(),( * txq

ntxp

q

.),(),( 00 xxptxp

3/79

Measurements

R )(:

)( 11

)( 22

)( NN

…

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Example:inexact ruler, …

Accuracy of measurements

exacti xXmaxxi

)(

We can calculate this number in controllable environments (in laboratory).

This error is not connected with probability.

accuracy

5/79

Inexact measurements

)](),([)(ˆ)(

)()( ,)()(

- accuracy of measurements

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Set-valued random variable

R )(ˆ:ˆ

)(ˆ 11

)(ˆ NN

)(ˆ 22

…

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Characteristics of discrete random variable

Mean value

iii

iii NNNE

)(1 ,)(1)(ˆ1)ˆ(

Variance

i

EN

Var i

2

)ˆ()(ˆ1)ˆ(

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Usually we don’t know probability density function (PDF)

Probabilistic methods require assumptions about the probability density function.

1)(2)()(,)()(

tXVartXEXVartXExP

t x

dxet 2

2

21)(

This formula is true only for Gauss PDF.

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Confidence interval})(:{1

P

)}(:{2

})(:{

PP

22

)(f

1

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Interval estimation of probability

RX :

x1,021,1

1 ,0)(

x

xxf X

)(XyY 3xy

y8,0

81,3

11 ,0

)('))(()(3 2

y y

y

yhyhfyf XY

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1 2 x5.1)( XE

1

08333.0))(()(2

1

2 dyxxxfXVar X

]78868.1 ,21132.1[])(,)([ XVarxXVarx

57735.0)()(

)(

dxxfXVarx

XVarxX 59313.0)(

)(

)(

dyyfYVary

YVaryY

Probability for X: Probability for Y:

)(xf X

12/79

,77739.1)( xyy 72261.5)( xyy

dxxfdyyfx

xX

y

yY

)(57735.0)(

x

y

x x

y

y

1

2

2

1

2

2

)(xyy

13/79

x

y

x x

y

y

2P

1P 3P

1P

2P

3P

)( xyy

Updating results using latest information

3

2

1

2

12

P

P

P

Olddata

Newdata

18/79

Properties of confidence intervals

1)Definition of confidence intervals is not based on the probability density function.

2) Confidence intervals can be defined using set-valued random variables

(uncertain measurements).

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Input membership function

x

3

3x

3x

2

2x

2x

1

1x

1x

0

0x

0x

)(xF

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000ˆ)ˆ(ˆ0 yxyx

111ˆ)ˆ(ˆ1 yxyx

222ˆ)ˆ(ˆ2 yxyx

333ˆ)ˆ(ˆ3 yxyx

)(yF

)(xyy ),(xF )]ˆ(),ˆ([)ˆ(ˆ xyxyxy

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Output membership function

0y

0y

0y

1y

1y

1

2y

2y

2

3y

3y

3)(yF

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Interval solutions of the slightly compressible flow equation

,...}ˆ,ˆ:,...),(inf{ ii pp

,...}ˆ,ˆ:,...),(sup{ ii pp

],[ˆ iiii pppp

Similar treatment for saturation.

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Example

Injection well

Production well

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Interval solution (time step 1)p_upper(t) - p_lower(t)

25/79

“Single-region problems”

xx 1 xx 2 xx 3

]2,1[x,321 xxxy

xxxx 321

xxxxxy 2321

2,

41]}2,1[:{ 2 xxx

x

y

1 2-1

1

2

xxy 2

41

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“Multi-region problems”

1x 2x 3x

5,4]}2,1[,,:{ 321321 xxxxxx

Solution of single-region

problem

Solution of multi-region

problem

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More constraints – less uncertainty

,321 xxxy

321 xxx constraints:

Result with constraints(single-region)

Results without constraints(multi-region)

2,

41 5,4

].2,1[ix

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Multi-region case

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Data filealpha_c 5.614583 /* volume conversion factor */beta_c 1.127 /* transmissibility conversion factor */

/* size of the block */

dx 100dy 100h 100

/* time steps */time_step 15number_of_timesteps 10

reservoir_size 20 20

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Interval solution (time step 5)

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Comparison

Single region - Multi-region

[0,55] [psi] [0, 390] [psi]

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Exact solution of equationswith interval parameters

2

1

2

1

2221

1211bb

xx

aaaa

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),( xkpp

**21

2 RLxx

sc ppkAxBqp

*Rpp *

Lpp x

],[

],,[

xxx kkk

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Monotone functions

1x 2x

)( 1xf

)( 2xf

0)(

dxxd f

)(}ˆ:)(sup{ xfxxxfy

)(}ˆ:)(inf{ xfxxxfy

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Extreme value of monotone functions

),...,,( 21 nxxxfy

nn xxxxxx ˆ,...,ˆ,ˆ 2211

nxxx ˆ...ˆˆˆ 21 x

)}ˆ(:)(min{ xxx Verticesyy

)}ˆ(:)(max{ xxx Verticesyy

n2 - calculations of y(x)

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Sensitivity analysis

If 0)( 0

xxf

, then )(),( xyyxyy

If 0)( 0

xxf

, then )(),( xyyxyy

),(xfy ].,[ xxx

]3,1[,2 xxy,2)( x

dxxdy

,422)2(

dxdy ,1)( xyy 9)( xyy

]9,1[ˆ y

37/79

Complexity of the algorithm, which is based on sensitivity analysis

),(xfy .xx

,1xf

,

2xf

nxf

… - n derivatives

),,...,,( 21 nxxxfy .,...,, 21

nxxxfy

We have to calculate the value of n+3 functions.

,......, ixf

00 ,..., ni xxf 1

n

,,1 ,..., nni xxfy

2

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Vector-valued functions nxxxyy ,...,, 2111

nxxxyy ,...,, 2122

nmm xxxyy ,...,, 21

…

In this case we have to repeat previous algorithm m times.We have to calculate the value of m*(n+2) functions.

39/79

Implicit function

)()( xQyxA

)()()( 1 xQxAxy

yxAxQyxAkkk xxx

)()()(

40/79

Sensitivity matrix

n

mmm

n

n

xy

xy

xy

xy

xy

xy

xy

xy

xy

...

............

...

...

21

2

2

2

1

2

1

2

1

1

1

xy

x 2y

2xy

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Sign vector matrix

mn

mm

n

n

SSS

SSSSSS

sign

...............

...

...

21

222

21

112

11

xy 2S

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Independent sign vectors

,ji SS .)1( ji SS

jijiji S *****

** )1(,, SSSSSS

Number of independent sign vectors:

],1[ m

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Complexity of the whole algorithm.

2*p – solutions (p times upper and lower bound).

],1[ mp

.21,12121 mnnpn

)()( xQyxA 1 - solution

n - derivatives .ix

yyxAxQyxA

kkk xxx

)()()(

)(xy

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All sensitivity vector can be calculated

in one system of equationsyxAxQyxA

kkk xxx

)()()(

yAQRHSkk

k xx

],...,[)( 1 nkx

RHSRHSyxA

Complexity of the algorithm: .22,12222 mp

kkx

RHSyxA )(

45/79

Sensitivity analysis method give us the extreme combination of the parameters We know which combination of upper

bound or lower bound generate the exact solution.

We can use these values in the design process.

min,min,1 ,..., nni xxfy max,max,1 ,..., n

ni xxfy

46/79

Example

,

111111111111

1111

4

3

2

1

4

3

2

1

QQQQ

yyyy

],2,1[ix

.

4321

321

43214321

4

3

2

1

xxxxxxx

xxxxxxxx

yyyy

,

2223

3222444

4321

4

4321

4321

4

3

2

1

xxxxx

xxxxxxxx

QQQQ

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Sensitivity matrix

1111011111111111

4

4

3

44

3

3

34

2

3

24

1

3

1

2

4

1

42

3

1

32

2

1

22

1

1

1

xy

xy

xy

xy

xy

xy

xy

xy

xy

xy

xy

xy

xy

xy

xy

xy

xy

x 1y

48/79

Sign vectors

4

4

3

44

3

3

34

2

3

24

1

3

1

2

4

1

42

3

1

32

2

1

22

1

1

1

xy

xy

xy

xy

xy

xy

xy

xy

xy

xy

xy

xy

xy

xy

xy

xy

signsignxyS

4

3

2

1

1111111111111111

1111011111111111

SSSS

S sign

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Independent sign vectors

1111

1111

2*

1**

S

SS

1111111111111111

4

3

2

1

SSSS

50/79

Lower bound- first sign vector

1111

)(

4

3

2

1

1*

xxxx

Sx

2344

))((

))((

))((

))((

))(())(())((

1*4

1*3

1*2

1*1

1*

11*

1*

Sx

Sx

Sx

Sx

SxQSxASxy

y

y

y

y

))(())(())(( 1*

1*

1* SxQSxySxA

]1,1,1,1[1* S]2,1[ix

51/79

Upper bound- first sign vector

2222

)(

4

3

2

1

1*

xxxx

Sx

4688

))((

))((

))((

))((

))(())(())((

1*4

1*3

1*2

1*1

1*

11*

1*

Sx

Sx

Sx

Sx

SxQSxASxy

y

y

y

y

]1,1,1,1[1* S

]2,1[ix

52/79

Lower bound – second sign vector

]1,1,1,1[2* S

2111

)(

4

3

2

1

2*

xxxx

Sx

1555

))((

))((

))((

))((

))(())(())((

2*4

2*3

2*2

2*1

2*

12*

2*

Sx

Sx

Sx

Sx

SxQSxASxy

y

y

y

y

]2,1[ix

53/79

Upper bound – second sign vector

1222

)(

4

3

2

1

2*

xxxx

Sx

5677

))((

))((

))((

))((

))(())(())((

2*4

2*3

2*2

2*1

2*

12*

2*

Sx

Sx

Sx

Sx

SxQSxASxy

y

y

y

y

]1,1,1,1[2* S

]2,1[ix

54/79

Interval solution

1344

))}(()),(()),(()),((min{ 2*

2*

1*

1* SxySxySxySxyy

5688

))}(()),(()),(()),((max{ 2*

2*

1*

1* SxySxySxySxyy

]5 ,1[]6 ,3[]8 ,4[]8 ,4[

y

55/79

Sensitivity in time-dependent problems

),(),( 1 hpQphpA ttt

11

),(),(),(

tt

k

t

kk

tt

hhhphpAhpQphpA

56/79

Sensitivity

57/79

Calculation of total rate and total oil production

PN

wi

wsciT tqtq

1)()(

NTS

iiiTP ttqN

1)(

wf

w

e

cwfscsc pp

srr

B

khppqq

21ln

20

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Interval total rate

PN

wwi

wsciT ptqtq

1),()(

PN

wwi

wsciT ptqtq

1),()(

59/79

Interval total oil production

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Exact value of total rate

PN

wi

wsciT tqtq

1)()(

PP N

w k

iwsc

N

wi

wsc

kiT

k hp

ptq

tqh

tqh 11

)()()(

)( iTR tqsign

hS

61/79

RShh R RShh R

),()( RiRi tt hpp ),()( RiRi tt hpp

))(,()( RiiTiT ttqtq p))(,()( RiiTiT ttqtq p

)](),([)(ˆ iTiTiT tqtqtq

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Truss structure example

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Accuracy of sensitivity analysis method (5% uncertainty)

Accuracy in %

0 1,04E-02

0 0,00E+00

0,003855 0,00E+00

0 0,00E+00

0 0,00E+00

0 0,00E+00

0 1,89E-03

0 5,64E-01

0,026326 0,00E+00

0 4,87E-03

0 1,21E-03

0 0

18 – interval parameters

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Taylor expansion method

m

iii

i

iii hh

huuu

10,

00

hhh

m

iii

i

iii hh

huuu

10,

00

hh

m

iii

i

iii hh

huuu

10,

00

hh

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Accuracy

%100,

,

exactimidi

iexactii uu

uudu %100

,

,

midiexacti

exactiii uu

uudu

66/79

Accuracy of two methods of calculation (20% uncertainty)

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Accuracy of two methods of calculation (50% uncertainty)

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Comparison 50% uncertainty Sensitivity method [%] Taylor method [%]    Comparison [%]

           

-0,03 -1,19 43,01 -48,34 143466,7 3962,185

-37,1 -0,39 -11,27 -46,95 69,62264 11938,46

-1,53 -0,24 28,41 -44,04 1956,863 18250

-0,25 -4,3 -41,91 21,75 16664 605,814

-0,29 -0,28 43,11 -47,35 14965,52 16810,71

-0,33 -0,04 -45,43 38,26 13666,67 95750

0 -1,97 31,88 -45,78 inf 2223,858

-13,59 -15,68 -32,33 -30,86 137,8955 96,81122

Si

SiTi

du

dudu

,

,, %100

Si

SiTi

du

dudu

,

,, %100Tidu ,

Sidu ,

Tidu ,

Sidu ,

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Time of calculation(endpoints combination method)

70/79

Time of calculation(First order sensitivity analysis)

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Time of calculation(First order Taylor expansion)

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Comparison

Number of interval parameters Sensitivity Taylor %

105 2 0,02 9900410 452 1,22 36949,18915 15 208 16,64 91294,23

Time in seconds

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APDL description N 1 0 0 N 2 1 0

MP 1 EX 210E9 F 3 FX 1000 R 1 0.0025

(description of the nodes)

(material characteristics)

(forces)

(other parameters – cross section)

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Interval extension of APDL language

MP EX 1 5 F 3 FX 5 R 1 10

(material characteristics)

(forces)

(other parameters – cross section)

Uncertainty in percent

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Web applications

http://zeus.polsl.gliwice.pl/~pownuk/interval_truss.htm

Endpoint combination method

Sensitivity analysis method

Taylor method

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Sensitivity analysis method

Pownuk A., Numerical solutions of fuzzy partial differential equation and its application in computational mechanics,

Fuzzy Partial Differential Equations and Relational Equations: Reservoir Characterization and Modeling (M. Nikravesh, L. Zadeh and V. Korotkikh, eds.), Studies in Fuzziness and Soft Computing,

Physica-Verlag, 2004, pp. 308-347

77/79

Monotonicity tests Taylor expansion of derivative Control of the gradient Interval methods

78/79

Conclusions In cases where data is limited and pdfs for

uncertain variables are unavailable, it is better to use imprecise probability (interval) rather than pure probabilistic methods.

Using interval methods we can create mathematical model of the reservoir which is based on very uncertain information.

79/79

Presented algorithm is efficient when compared to other methods which model uncertainty, and can be applied to nonlinear problems of reservoirs simulations.

Sensitivity analysis method gives very accurate results.

Taylor expansion method is more efficient than sensitivity analysis method but less accurate.