7.Effective medium Upscaling problem Backus averaging O’Doherty-Anstey approximation Reuss and...

7.Effective medium

• Upscaling problem

• Backus averaging

• O’Doherty-Anstey approximation

• Reuss and Voigt models

• Bio-Gassmann model

• Hertz-Mindlin model

Upscaling problem

Does seismic wave see thethin-layering?

Upscaling problem

• From microscopic to macroscopic scale

• From pore (graine) scale (millimeters)• From log-scale (centimeteers)

Upscaling problem

• Traditionally, upscaling has meant upscaling of reservoir petrophysical properties and flow parameters dedicated for reservoir fluid flow simulation. However, due to the progresses mentioned above, there is a need to extend the concept of upscaling of geological models, for rock physics properties, seismic modelling and analysis. For instance, in 4D history matching, the need for up and downscaling might differ from the traditional concept of upscaling.

Backus averaging

• Sequential Backus Averaging is a method of averaging the properties of a stack of thin layers so they are similar to average properties of a single thick layer.

Figure 7.1. The Backus averaging scheme

Backus averaging• The advantage of Sequential

Backus Averaging is that no artificial "blocks" are introduced into the geology during the upscaling of the well-log data. In this example the density log is blocky, but the compressional- and shear-wave velocity logs have gradational tops and appear thicker. Blocking would distort the amplitudes. Furthermore, if blocking were based solely upon either the density or the sonic curves, the result would be wrong for the other curve.

Figure 7.2. The Backus averaging versus blocking averaging

Backus averaging

• Thin beds appear thinner at oblique incidence angles.

Figure 7.3. The thin beds

Backus averaging• At nonnormal incidence, the averaging operator must be adjusted to include the

apparent bed thinning.

Figure 7.4. Adjusting of averaging operator

Backus averaging

• The offset synthetic shows differing AVO signatures for the same elastic property contrasts, associated with step-functions, blocky beds, and gradational interfaces.

Figure 7.5. AVO signatures from different models

Backus averaging

ij kC , , k 1,M

ijˆ ˆC ,

(7.1)

Backus averaging

133

21311

213313

13313

133

cccppcc

pccc

A

1

44cp

pB

How many combinations of the stiffness coefficients enter these matrices?

(7.2)

(7.3)

Backus averaging

2 11 11 13 33

12 33

13 13 33

14 44

M

k kk 1

A c c c

A c

A c c

A c

1m d m

D

23

11 12

313

2

332

444

Ac A

A

Ac

A

1c

A

1c

A

(7.5)(7.4)

The effective vertical velocity from Backus averaging

2 2 21 1

1

2 21 1

21

2 2 21 1

1 1

1 12

1 4

1

N Nj

j jj jEF j j

N Nj k j j k k

j k jTA j k k k j j

N Nj k jk

j k jTA j k jk

dd

V D v

d d v v

V D v v v v

d d r

V D v v r

Stovas and Arntsen, 2003

(7.6)

Layering

0,00 0,05 0,10 0,15 0,20 0,25 0,30 0,35 0,40

-0,04

-0,02

0,00

0,02

0,04

M1

Time, s

-0,04

-0,02

0,00

0,02

0,04

M2

-0,04

-0,02

0,00

0,02

0,04

M4

-0,04

-0,02

0,00

0,02

0,04

M8

Figure 7.6. The layering effect (each model computed by compression and doubling of the previous one)

Reflection-transmission versus layering and contrast

0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0-2

0

2

M2

M1

Time, s

-2

0

2-2

0

2

M4

-2

0

2

M32

M28

M24

M20

M16

M12

M8

-2

0

2-2

0

2-2

0

2-2

0

2-2

0

2-2

0

2

0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0-1

0

1

M2

M1

Time, s

-0,5

0,0

0,5

-0,5

0,0

0,5

M4

-0,2

0,0

0,2

M32

M28

M24

M20

M16

M12

M8

-0,1

0,0

0,1

-0,1

0,0

0,1-0,03

0,00

0,03

-0,01

0,00

0,01

-0,005

0,000

0,005-0,003

0,000

0,003

0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0-0,5

0,0

0,5

M2

M1

Time, s

-0,5

0,0

0,5-0,5

0,0

0,5

M4

-0,5

0,0

0,5

M32

M28

M24

M20

M16

M12

M8

-0,5

0,0

0,5-0,5

0,0

0,5-0,5

0,0

0,5-0,5

0,0

0,5-0,5

0,0

0,5-0,5

0,0

0,5

0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0

-1

0

1

M2

M1

Time, s

-1

0

1

-1

0

1

M4

-1

0

1

M32

M28

M24

M20

M16

M12

M8

-1

0

1

-1

0

1

-1

0

1

-1

0

1

-1

0

1

-1

0

1

0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0-2

0

2

M2

M1

Time, s

-2

0

2-2

0

2

M4

-2

0

2

M32

M28

M24

M20

M16

M12

M8

-2

0

2-2

0

2-2

0

2-2

0

2-2

0

2-2

0

2

0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0-1

0

1

M2

M1

Time, s

-0,5

0,0

0,5

-0,5

0,0

0,5

M4

-0,2

0,0

0,2

M32

M28

M24

M20

M16

M12

M8

-0,1

0,0

0,1

-0,1

0,0

0,1-0,03

0,00

0,03

-0,01

0,00

0,01

-0,005

0,000

0,005-0,003

0,000

0,003

0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0-0,5

0,0

0,5

M2

M1

Time, s

-0,5

0,0

0,5-0,5

0,0

0,5

M4

-0,5

0,0

0,5

M32

M28

M24

M20

M16

M12

M8

-0,5

0,0

0,5-0,5

0,0

0,5-0,5

0,0

0,5-0,5

0,0

0,5-0,5

0,0

0,5-0,5

0,0

0,5

0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0

-1

0

1

M2

M1

Time, s

-1

0

1

-1

0

1

M4

-1

0

1

M32

M28

M24

M20

M16

M12

M8

-1

0

1

-1

0

1

-1

0

1

-1

0

1

-1

0

1

-1

0

1

Figure 7.7. The reflection (bottom)and transmission (top) responses withdifferent contrasts (to the right is 4 times larger).

Binary medium (multiples)

0,0 0,1 0,2 0,3 0,4 0,5

0M1

Time, s

0M2

0M3

0M4

0M5

0M6

0

Full reflected field

M7

0,0 0,1 0,2 0,3 0,4 0,5

0M1

Time, s

0M2

0M3

0M4

0M5

0M6

0

Primaries only

M7

0,0 0,1 0,2 0,3 0,4 0,5

0M1

Time, s

0M2

0M3

0M4

0M5

0M6

0

Multiples only

M7

Figure 7.8. Multiples contribution into the reflection response

Propagation versus contrast

0,00 0,05 0,10 0,15 0,20 0,25 0,30 0,35 0,40 0,45 0,50

Time, s

r=0.87

second multiple

first multiple

Model: 256 x 1m

primary transmission

r=0.79

r=0.70

r=0.60

r=0.48

r=0.33

r=0.16

r=0.00

Figure 7.9. Transmission from thin layer model (change in r due to change in only)

Effective properties versus net-to-gross

0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0

2,10

2,12

2,14

2,16

2,18

2,20

linear

, g/cm3

Net-to gross ratio

0,91,01,11,21,31,41,5

non-linear

VS0

, km/s

2,00

2,05

2,10

2,15

2,20

2,25

non-linearV

P0, km/s

0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0-0,08

-0,06

-0,04

-0,02

0,00

symmetric

Net-to gross ratio

-0,14

-0,12

-0,10

-0,08

-0,06

-0,04

-0,02

0,00

non-symmetric

Figure 7.10. Effective properties from Backus averaging in a binary medium

Stovas, Landro and Avseth, 2004

Turbidite sequence from Ainsa basin

Figure 7.11. Turbidite system as an example of binary medium

Binary medium

1 2

1 2 * *12 21

1 10 01

1 10 0

Si i

i i

r r a be e

r r b at t e e

1 2 2

1 2 2

1

22

2

2

22 2

1

1

1 2 sin

1 1

i i

i i

i

e r ea

r

re e irb e

r r

2 k k kfd v

(7.7)

(7.8)

(7.9)

Binary medium

2

1 2 1 22

2

1 2 1 22

2Re cos sin sin

1

2Im sin cos sin

1

ra

r

ra

r

2 2det 1 S a b

2 2

2 2

22 2 *2

222 2 2* 2

2

2 sin2 4 sin

2 sin2 1

S S I I

i i

H

i i

r i r e ea b a b r

i r e e rab a b r

(7.10)

(7.11)

(7.12)

The propagator matrix is not unitary

Binary medium

det 0 S I

21,2 Re 1 Re a i a

From the characteristic equation

we compute the eigenvalues

(7.13)

(7.14)

Binary medium

21,2 Re Re 1

, Re 1

, Re 1

i

a a

e a

e a

The propagating regime with complex eigenvalues and the blocking regime with real eigenvalues:

(7.15)

Propagating and blocking regimes

0 100 200 300 400 500 600 700-1

0

1

Frequency, Hz

-1

0

1-1

0

1-1

0

1-1

0

1-1

0

1-1

0

1

M1

M2

M4

M8

M16

M32

M64

r=0.87

0 100 200 300 400 500 600 700-1

0

1

Frequency, Hz

-1

0

1-1

0

1-1

0

1-1

0

1-1

0

1-1

0

1

M1

M2

M4

M8

M16

M32

M64

r=0.16

Figure 7.12. Re a as a function of frequency versus layering and contrast.Filled low frequency area relates to an effective medium, next coming gaprelates to transition medium. The interchanging of these zones is repeatable.

Velocity limits

1 2

1 2

TA

dv

d dv v

21 2

2 2 2 21 2

1 1 4 1

1

EF TA

d d r

v v d v v r

1 2

1 22 d v v

vd d

The time average limit means thatthe pulse width is much less than the propagation time through the cycle)

The effective medium limit can be computed assuming phases beingsmall (low frequency limit)

The geometrical average limit

(7.16)

(7.17)

(7.18)

Velocity limits versus volume fraction

0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,01000

2000

3000

4000

5000

6000

7000

8000

r=0.16r=0.48

r=0.87

vRT

v

Vel

oci

ty, m

/s

Volume fraction

vRT

v v

EF

Figure 7.13. Velocity versus fraction. The larger reflection coefficient the more deviationbetween time-average and effective medium velocities. The position for maximum difference between them moving to high values of volume fraction with r increase.

Stack of binary layers

1S UΛU

1 2, Λ diag

1 2

1 1

Ua b a b

The propagator matrix can be represented by the eigenvalue decomposition

(7.21)

(7.20)

(7.19)


1 22 2 21 2 11

22 21 21 22 2 1 2 22 1 21

1

Q S UΛ UM M M M

M M

M M M M

u u

u v u u u u

1 2 122

2 2 1 1

2 1112 22

2 2 1 1

D M M

M M

D M M

t qa a

br q q

a a

(7.23)

(7.22)

Product of M cycles

Transmission and reflection response


cos Re a

Propagating regime

Re 1a

1,2 ie

Blocking regime

Re 1a

1,2 e

cosh Re a

(7.24)

(7.25)

(7.26)


2

coscos

1sin

sin

M

CM

C b

1

2

2

2

sin

sin cos Im sin 1

sin

sin cos Im sin 1

i

D

i

D

et

M i a M C

b M Cer

M i a M C

Propagating regime

(7.28)

(7.27)

Stack of binary layersBlocking regime

1

2

2

2

sinh

sinh cosh Im sinh 1

sinh

sinh cosh Im sinh 1

i

D

i

D

et

M i a M C

b M Cer

M i a M C

2

coshcos

1sinh

sinh

M

CM

C b(7.30)

(7.29)


0 100 200 300 400 500 600 700-1,0-0,50,00,51,0

Frequency, Hz

-1,0-0,50,00,51,0

-1,0-0,50,00,51,0

-1,0-0,50,00,51,0

-1,0-0,50,00,51,0

-1,0-0,50,00,51,0

-1,0-0,50,00,51,0

M1

M2

M4

M8

M16

M32

M64

r=0.87

0 100 200 300 400 500 600 700-1,0-0,50,00,51,0

Frequency, Hz

-1,0-0,50,00,51,0

-1,0-0,50,00,51,0

-1,0-0,50,00,51,0

-1,0-0,50,00,51,0

-1,0-0,50,00,51,0

-1,0-0,50,00,51,0

M1

M2

M4

M8

M16

M32

M64

r=0.16

Figure 7.14. cos as a function of frequency versus layering and contrast(blue line is for the reference time average medium).

Stovas and Ursin, 2005


0 100 200 300 400 500 600 700-20-10

01020

Frequency, Hz

-20-10

01020

-20-10

01020

-20-10

01020

-20-10

01020

-20-10

01020

-20-10

01020

M1

M2

M4

M8

M16

M32

M64

r=0.87

0 100 200 300 400 500 600 700-2-1012

Frequency, Hz

-2-1012

-2-1012

-2-1012

-2-1012

-2-1012

-2-1012

M1

M2

M4

M8

M16

M32

M64

r=0.16

Figure 7.15. Amplitude C as a function of frequency versus layering and contrast(the gaps relates to the extremely large values).


0,00 0,02 0,04 0,06 0,08 0,10-101 TEM

TRT

M4

M2

M1

r = 0.87

Time, s

-101

-101

-101

M64

M32

M16

M8

-101

-101

-101

0,000 0,013 0,026 0,039 0,052 0,065 0,078 0,091-404 TEM

TRT

M4

M2

M1

r = 0.16

Time, s

-404

-404

-404

M64

M32

M16

M8

-404

-404

-404

Figure 7.16. Transmission response versus layering and contrast.Note the difference between TRT (transmission time for time average medium) andTEM (transmission time for effective medium). Weak transmission for r=0.87 andModel M16 is due to the wavelet spectrum is in the blocking regime, see Figure 7.12)


0,00 0,02 0,04 0,06 0,08 0,10-404

Time, s

-404

M1

M2

M4

M8

M16

M32

M64

-404

-404

2*TEM

-404

-404

-404

r = 0.87

0,00 0,02 0,04 0,06 0,08 0,10-1,50,01,5 2*TEM

Time, s

-1,50,01,5

M1

M2

M4

M8

M16

M32

M64

-1,50,01,5

-1,50,01,5

-1,50,01,5

-1,50,01,5

-1,50,01,5

r = 0.16

Figure 7.17. Reflection response versus layering and contrast.


0 100 200 300 400 500 600 7000,00,51,0

M64

M32

M16

M8

M4

M2

M1

r=0.87

Frequency, Hz

0,00,51,0

0,00,51,0

0,00,51,0

0,00,51,0

0,00,51,0

0,00,51,0

0 100 200 300 400 500 600 7000,00,51,0

M64

M32

M16

M8

M4

M2

M1

r=0.16

Frequency, Hz

0,00,51,0

0,00,51,0

0,00,51,0

0,00,51,0

0,00,51,0

0,00,51,0

Figure 7.18. Transmission (solid line) and reflection (dotted line) amplitudes as a function of frequency versus layering and contrast.

Phase velocity

0 100 200 300 400 500 600 700

1500

3000

4500

6000

M64

M32

M16

M8M4

M2

M1

VEF

VTA

r=0.87

Frequency, Hz

0 100 200 300 400 500 600 7003600

3800

4000

4200

M64

M32

M16M8

M4

M2M1

VEF

VTA

r=0.16

Frequency, Hz

Figure 7.19. Phase velocity as a function of frequency versus layering and contrast.The effective medium is the low frequency part (around effective medium limit), thetransition medium is for dramatical increase in velocity and time average medium isfor oscillating part around time average velocity limit. Note that for small r, the widthof transition zone is narrow comparing with high r case.

Transition from effective to time average medium

• Critical wavelength-spacing ratio:

/d=3 (Helbig, 1984)

/d=5-8 (Carcione et al., 1991)

/d=10 (Marion et al., 1992, 1994)


Re 1a

1 21

tan tan2 2 1

r

r

1

1

1

2

1tan

1

1tan

2 1

ra

d r

ra

d r

(7.31)

(7.34)

(7.33)

(7.32)


0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9

0

4

8

12

16

20

24

28

32

36

40/d

Time average medium

Effective medium

Transition zone

Absolute value of r

Figure 7.20. Effective, transition and time average medium (volume fraction 0.5)versus contrast.

O’Doherty-Anstey approximation

• Plane waves are normally incident on a sequency of horizontal layers. If the layers are lossless the shape of the frequency spectrum of the reflection response depends on the reflection coefficient series. The law of dependence can be found by solving the wave equation for the boundary and initial conditions of the seismic experiment. The O’Doherty-Anstey formula is an approximation to this law, and its validity would imply a lowpass spectrum of the reflection/transmission response if the reflectivity power spectrum has a highpass trend.


The ODA result for the retarded transmissivity caused by propagation through a set of layers is:

where N is the number of layers and R+(z) is the causal half of the normalized autocorrelation of the reflectivity function in a z-transform notation

z-transform:

(7.35)

(7.36)iz e


its Fourier representation

(7.37)

(7.38)


Now recall that reflectivity is a differential process, and if the elastic parameters are stationary in time, then

and our first, scaling, coefficient goes to zero leaving,

(7.40)

(7.39)


Figure 7.21. Examples of submillimetric fine layering from Beringen coal mine:Top – coarse sedimentary rock (sandstone),Bottom – fine sedimentary rock (shaly siltstone)


Figure 7.22. Thin micrograph of Rotliegend Sandstone (at 2990m depth).Left – laminated structure due to differences in grain size and packing.Right – details of two laminae, upper: coarser grained laminae with intergranular pores, lower: finer grained laminae with partly filled inrergranular space by detrital clays and dolomite.


Figure 7.23. Thin section micrographs. Scale = 0.25 mm. (A) Single lamina of very fine-grained, poorly sorted quarz sandstone in shale (2570m depth).(B) Two laminae of very fine-grained, well sorted quartz arenit interlaminated wirh sandy shale

(2920m depth).(C) Laminated, very fine grained sandstone and interbedded silty shale (2650m depth)


Figure 7.24. Thin section of Rotliegend sandstone (left) and P-wave increase with triaxial pressure increase


2 2

0 0Dkk k DPSDkk

1 1t , k, z exp i z exp r z r z

2 2

0

z''

z'

z

g12

g12

g11

g11

S

P

PP

P

P

PPP

P

P

P

Pg

12

g12

g22

g22

g12

g12

g11

g11

S

SS

SS S

SS

SSP

P

P

P

P

PPPP

z

z'

z''

z'''

0

(7.40)

Stovas and Ursin, 2004

Figure 7.25. Contrubution of first-order multiples into PP transmission (left) and PS reflection (right).


* *1

Q QN

N N

N jjN N

A B

B A

1

2

1 1

1

1 ...1

Nj k

i N Ni

N k jNk j k

jj

eA r r e

r

2

1

1

...

1

N jN

Nii

jj

N N

jj

e r e

Br 1

k

k jj

The propagator matrix for the stack of N layers (see eq. 7.7)

j j jd v(7.44)

(7.43)

(7.41)

(7.42)

Stovas and Arntsen, 2003


2 2

1 1

1det det

1

Q QN N

jN N N j

j j j

rA B

r

det 1QN

Determinant of propagator matrix

For binary medium

(7.46)

(7.45)


The elements of the total propagator matrix

12

1 1

1

1,1 1 ...1

Nj k

i N Ni

N k jNk j k

jj

eQ r r e

r

2

1

1

...

1, 21

N jN

Nii

jj

N N

jj

e r e

Qr

(7.48)

(7.47)


1 1

12

1 1

12,2

1 ...

N

D

j k

Ni

kN k

N N Ni

k jk j k

e rt Q

r r e

22

11

12

1 1

...

1, 2 2,2

1 ...

N jN

D

j k

Nii

jjN

N N N Ni

k jk j k

e r e

r Q Q

r r e

The transmission and reflection response:

(7.50)

(7.49)


1

1

1

D

N

kN k

rt

1

1

N

kk

r

11

The transmission amplitude

consists of two two terms:

attenuation due to transmission

attenuation due to scattering

(7.51)


The transmission phase

Imtan

1 Re

N N a

also consists of two terms

the time-average term

the scattering term

N

Imtan

1 Rea

(7.52)


1

1 1

1

1 1

1 1 Im 1 1 Imtan tan

Re 1 Re

sin 2 ...1 1

tan1 cos 2 ...

D

D TA

N N

k j j kk j k

N NTA

k j j kk j k

ta a

V D t V D

r r

aV D r r

The phase velocity

(7.53)

1

1 1

100

1 1

...1 1 1 2

lim1 ...

N N

k j j kk j k

N NTA

k jk j k

r r

V V V D r r

The zero-frequency limit

(7.54)


1 e

1

1 1

cos2

1

1

N N

k j j kk j k

r r NND k

k

t e r

1

1 1

2

0

N N

k j j kN k j k

r r

TAV V e

With approximation of the type

and transmission amplitude 7.51 we obtain

and zero-frequency limit 7.54 becomes

(7.57)

(7.56)

(7.55)


0 5 10 15 20 25 30 350,5

0,6

0,7

0,8

0,9

1,0

Tra

nsm

iss

ion

Frequency, Hz

0 5 10 15 20 25 30 351980

2000

2020

2040

2060

2080

2100

2120

data, = 2 weak-contrast O'Doherty-Anstey

Vel

oci

ty, m

/s

Figure 7.26. The phase velocity and transmission amplitude versus frequency. The comparison between exact, weak-contrast and O’Doherty-Anstey approximation. Note the low frequency range, less than 5 Hz.

Reuss model

• Isostress model (valid for suspensions, with the fluid phase load-bearing), porosity is greater than critical porosity.

Ni

i 1R i

1 1 d

M D M

The critical porosity separates the mechanical and acoustic behavior into two disctinct domains. For porosity less the critical one the mineral grains are load-bearing.For porosity larger the critical one the sediment becomes a suspension.

(7.58)

Voigt model

• Isostrain model (the load-bearing domain), porosity is less than critical porosity

N

V i ii 1

1M d M

D

(7.59)

Thin-layer model

Figure 7.27. Snapshot for a thin layer model (f=30Hz)

Reuss averaging

Figure 7.28. Snapshot for an effective Reuss model (f=30Hz)

Voigt averaging

Figure 7.29. Snapshot for an effective Voigt model (f=30Hz)

Average slowness

Figure 7.30. Snapshot for an effective average slowness model (f=30Hz)

Average velocity

Figure 7.31. Snapshot for an effective average velocity model (f=30Hz)

Backus average

Figure 7.32. Snapshot for an effective Backus model (f=30Hz)

Bio-Gassmann model

• Biot (1956): frequency dependent velocities of saturated rocks in terms of the dry rock properties

• Gassmann (1951): the low frequency limit of Biot equations

Assumptions and limitations:- Rock is isotropic- All minerals making up rock have same bulk and

shear moduli- Fluid-bearing rock is completely saturated

Biot equations can be extended to VTI medium

Gassmann model

2fr ma2fr

ma frma

f ma

K K1 4K

3 K KK 1

K K

2

frK

maK

fK

P-wave velocityS-wave velocityDensityBulk modulus of solid frameworkShear wave modulusIntrinsic modulus of solid matrixSaturated fluid bulk modulusPorosity

(7.60)

(7.61)

Fluid densityMatrix densityOil densityWater densityOil bulk modulusWater nulk modulus

Gassmann model f ma1

f w o

1 S 1 S

K K K

f

ma

f w oS 1 S

wK

w

oK

o

Density

Fluid bulk modulus

Fluid density (7.64)

(7.63)

(7.62)

Hertz-Mindlin model

• The Hertz-Mindlin model (Mindlin, 1949) can be used to describe the properties of precompacted granular rocks

Poisson’s ratioShear modulusPorosityAverage number of contacts per grainHydrostatic confinig pressure

Hertz-Mindlin model

22 2

3eff 22

C 1K P

18 1

22 2

3eff 22

3C 15 4P

5 2 2 1

C 9

P

(7.66)

(7.65)

Gassmann-Mindlin

Figure 7.32. The vertical P-wave and S-wave velocities versus water saturation and effective pressure changes.

Stovas and Landro, 2005

Gassmann-Mindlin

Figure 7.33. Relative (to the initial model) changes in P-wave velocity, S-wave velocityand density versus water saturation and effective pressure changes. Within the Hertz-Mindlin model density does not change with pressure.

Gassmann-Mindlin

Figure 7.34. The behavior of the PP and PS reflection coefficients with changing water saturation. The initial model reflection coefficients are plotted by circles. The curves are sampled in the water saturation change of 0.2.

Gassmann-Mindlin

Figure 7.35. The behavior of the PP and PS reflection coefficients with changing effective pressure. The initial model reflection coefficients are plotted by circles. The curves are sampled in the change in effective pressure of 0.001 Gpa.

Gassmann-Mindlin

Figure 7.36. Stacked PP reflection coefficient versus saturation and pressure

Gassmann-Mindlin

Figure 7.36. Stacked PS reflection coefficient versus saturation and pressure

7.Effective medium Upscaling problem Backus averaging O’Doherty-Anstey approximation Reuss and...

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Transcript of 7.Effective medium Upscaling problem Backus averaging O’Doherty-Anstey approximation Reuss and...