project 2 paper

8/12/2019 Project 2 Paper

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Inversion of permeability and porosity from transient pressure data.

Project Assigment:

Given the following reservoir and well data:

Two wells, 360 feet apart, have been completed in a new reservoir and then shut in. Thesize of the reservoir is not yet nown but all point to a large reservoir. !ell "0. # wasproduced at $#% &T'() while well remained shut in the pressure data recorded at well"0. * together with other pertinent data are given below.

+easured 'ottomhole resssure-psia

Time-hrs ressure-psia0.0 3/%3

0.033 3/%30.0611 3/%3

0.#000 3/%30.#330 3/%30.#6/0 3/%30.*$00 3/%30.$000 3/%30./$00 3/%*#.0 3/%#*.0 3/113.0 3/16 $.0 3/1#0.0 3//%

*.0 3//1.0 3//0/*.0 3/6/ %6.0 3/6$

2eservoir and well datao0.%*'o#.06 2'(&T'4#3ftrw0.*/ ftc

t#35#0

6

psi

#

7t is re8uired to determine the formation permeability and porosity using inversionapproach. The true value are given, 93# md and 0.*3/


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7 art.ssuming a priori information of permeability and porosity with Gaussian distribution.;se log9 and parameter.

!ith a mean0.*1, log 9meanlog-$00*./ , 0.0$ and log90.0$ a normal

distribution was generated for each of those parameter as is shown in


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77 part a. **5** +atri> allow us to calculate the value for the prior function. ?>>T

and the prior term of the function is given by.

( ) ( )( )2

loglog)(

22

log

+

=

9

99>p

which could be e>pressed in this case as follow

( ) ( ) ( )3)(2

1exp

2

1

21

2/

>p

?

>

n

=

where

)1.3(2log2

1

== nand? >


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where,

( )1.4)(1 dtt

e>@

>

t

=

)2.4(000264.02

wto

)rc

9tt

=

( )3.42.142

)),((

'8

tr,,9h,)

=

( )4.4w

)r

rr =


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#, >*BBBBBB., >#/C. The drainage radii- r is considered to be the distancebetween the two analyzed wells.

The lielihood function is given by

( ) ( ) ( )5)()()/( 1 mgd?mgd>dy


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( )6

1...........................00

0...........................10

0...........................01

The lielihood term is reduced to.

( ) ( )7)()/(2

1

=

==n

i

mgd>dyp

where,

dobserved and g-m calculated using e8. 3.

This lielihood function is then calculated obtained

( ) ( ) ( )8)/(2

1exp

2

1

21

2

>dyp

?

>

n=

=


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The e8uations for that are the following

( ) ( ) ( ) ( ) ( )9)()()/( 11 +== >?>mgd?mgddy>T

d

T

Thus, the posterior term could be e>pressed as.

( ) ( ) ( )

( )10loglog

)()/(

22

log

2

1

+

+==

=

9

n

i

99mgddy>p

The posterior function could be defined through 'ayes Theorem as.

( )11)(*)/()/( >p>dypdy>p ===

;sing the **5** matri> already mentioned the respective value of the posterior is thencalculated fi>ing one value of the porosity and changing permeability. This allow us tofound the surface in which the pea should be the true given value. To find this value aoptimization process was also used.

To be able to do this procedure 7 calculated the partial derivative of respect to log 9and respectively. ?ombining e8s. 3 to 3. the following e8s. are obtained.

( )122

3.2

log

2

2

12

1

=

t9

?

9et9

?

@9

9?,

and

( )132

2

1

=

t9

?

e9

?,

where,

000264.0*4;

2.142 2

21

t?r?h

'8?

==


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The sensitivity matri> is given by

( )14

log

log

log

1717

22

11

,,

,,

,,

9

9

9

Then for an initial guess of log9 and and using e8s. 3 to 3. we obtain the -dg-m

matri>.

( )15

1717

22

11

calobs

calobs

calobs

,,

,,

,,


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>T>dT >>?mgd?Gd>?G?G =+

Then >new>oldDd>

This result is shown in Table #.

The value found by this and the graph way is the so call ma>imum a posterior and isgoing to be used for the ne>t stages of the proEect.

imum lielihood estimatewith porosity0.** and 9#6.%.

0.

060

0.

120

0.

180

0.

240

0.

300

0.

360

0.4

20

0.

480

2.

500

2.

560

2.

620

2.

680

2.

740

2.

80

0

2.

860

2.

920

0

5

10

15

20

25

30

log(k)

Likelihood

0.

06

0

0.1

20

0

.180

0.

240

0.

300

0.

360

0.

420

0.

480

2.

500

2.5

60

2.

620

2.

680

2.

740

2.

800

2.

860

2.

920

0

50

100

150

200

250

300

log(k)

Posterior


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imum a posterior estimatewith porosity0.** and 936.$.

Table #. &chematic procedure to calculate the ma>imum a posterior estimate from theoptimization.

7n figure $ we could observe the surface for the posterior function with the ma>imum aposterior value.

777 art.fter determine the most liely 9 and -+ from the posterior distribution, thesensitivity matri> G-> e8. # was calculated for each pair of log 9and varying oneparameter and fi>ing the other one. Those values are shown below.

Hy= GTCd

-1G Cx

-1Hy+Cx

-1

7.286286E+03 5498.958487 408.3770462 0 7.694663E+03 5498.958487

5498.958487 6073.891983 0 399.0213866 5498.958487 6472.91337

d-g() x-x!

0.000000 -0.078463

-0.000006 -0.057078

-0.000297

-0.002684 GTCd

-1(d-g()) Cx

-1("-"!)

-0.010877 -31.334434 -32.042613

-0.072659 -23.735427 -22.775448

-0.598940

-0.347500 GTCd

-1(d-g())-Cx-1("-"!)

-0.118287 0.708179

0.255425 -0.959979

0.306668

-0.454416

0.440525

-0.068971

-0.557712

-0.210197

-0.099645

7.694663E+03 5498.958487 #"1 0.708179

5498.958487 6472.91337#"$

-0.959979

#"1= 0.000504022

#"$= -0.00057649

K 417.0128511

Poro 0.223700728

% &ter 4

=

'ensitiity *trix - G

-3.343931E-09 6.929489E-09

-2.856312E-04 6.166587E-04

-7.868064E-03 1.766027E-02

-4.736913E-02 1.107007E-01

-1.410967E-01 3.435199E-01

-5.566259E-01 1.495057E+00

-1.819767E+00 6.564644E+00

-2.167776E+00 1.074968E+01

-1.903498E+00 1.375586E+01

9.468346E-01 1.991252E+01

3.994121E+00 2.252539E+01

8.962905E+00 2.486047E+01

1.709458E+01 2.676930E+01

2.866602E+01 2.794976E+01

3.834478E+01 2.838385E+014.411666E+01 2.853004E+01

4.824280E+01 2.860342E+01


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The 4essian associated with the data points 4y could be calculated through the relation

( )171

G?GFF4 dTT

yy

==

Then the rescaled sensitivity matri> 4essian y is given by.

( )18 21

21

>y>y ?4?4 =

To calculate the eigenvalues of y the determinant is e8ualized to zero.

190

0

2212

1211=

=

44

444y

fter solving the 8uadratic e8uation coming from this e8uation we obtained# and *.


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>

44

44

Thus, the general e8uation for the eigenvector is.

(21

1

)/( 11122

=

44>Hi

The rescaled 4essian matri> IS.

( )22 744 y+=


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7n and the eigenvalues of the rescaled 4essian matri> are shown.

and the eigenvalues of the rescaled 4essian matri>.

The analysis of y and tell us that in this case the data from the pressure analysis have

a strong influence in the posterior distribution.

7H. artThe original 4essian from the following relation

( )2311 += >d

T?G?G4

Hy= G!"

-1G Cx

+,Cx

+,Hy Hy=Cx

+,HyCx

+,

6.967084E+03 5255.075962 0.049484517 0 344.7628062 260.0448973 17.06042104

5255.075962 5807.731023 0 0.050061276 263.0758071 290.7424245 13.01817932

eigen(*lue eigen(ector

y1 2.729362623 1 = 0.22 -9.083892E-01 2 = 0.22 1.100850E+00

y2 28.88599512 1 1

Cx-1

408.3770462 0

0 399.0213866

. .

H = #y+ $

18 13.01817932

13.01817932 15.5549367

_

eigen(*lue eigen(ector H

1 3.729362623 1 = 0.22 -0.908389244 2 = 0.22 1.100849671

2 29.88599512 1 1


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fter found the original 4essian the covariance of the a posterior distribution is given by

( )241= 4?>

imation after used the pressure data.

round the ma>imum a posterior p.d.f appro>imates a Gauusian with >and ?> 4 #This appro>imation is valid regarless of the actual p.d.f using in the prior distribution.>HT =

The calculated transposes eigenvector of 4 is

( )2766173.7497.

7497.66173.=

T

H

and ->> is the matri> given by

( )28.

loglog)(+:,

9+:,9>>

=


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project 2 paper

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