Biot, Gassmann and me

Brian Russell

Hampson-Russell, A CGGVeritas Company

Introduction

In this talk, I want to discuss how the work of Biot and

Gassmann has shaped both Hampson-Russell’s software

and my own scientific development over the last 27 years.

I will start with a discussion of Gassmann’s work, and how it

influenced version 1.0 of AVO in 1985 (at Veritas!).

I will then discuss the importance of Biot’s work, which I first

appreciated while working to generalize the LMR equation.

This became even clearer while working with Dave Gray to

generalize his two three-term AVO expressions.

Gassmann’s contribution to the effects of porosity then lead

me to develop a new rock physics template.

Finally, I will look to the future with a discussion of

Gassmann in anisotropic rocks.

2

Poroelasticity

Rock mineral grains Pores and fluid

Dry rock

frame, or

skeleton

(pores empty)

Saturated

Rock

(pores full)

Biot (1941) and Gassmann (1951) were the founders of

poroelasticity theory, which they developed in quite different

ways by comparing the compressibility of a dry and saturated

volume of rock, as shown below:

3

Pressure and compressibility

By considering the pressure effects on samples of a dry

and saturated rock, I will show how Gassmann’s result can

be derived.

The compressibility of the rock, C, which is the inverse of

the bulk modulus K, is the change of the volume of the rock

with respect to pressure, divided by the volume:

pressure. volume, : where,11

PV

dP

dV

VKC

In the above equation, there are two fundamental types of

pressure: confining pressure, PC, and pore pressure, PP.

Also, there are three different volumes to consider: the

volume of the bulk rock, the mineral and the pore space.

Three models of a porous rock

Utilizing these concepts, we can build three simple models of

the rock volume, as shown here (Mavko and Mukerji, 1995):

DPc

DPc

DPc

DPc

DPc DPc DPc

DPc

DPc

DPc

DPc DPp

A. Mineral case B. Dry case C. Saturated case

In A, we compress the mineral, in B we compress the mineral

and dry pore, and in C the mineral and saturated pore.

Mineral Dry Pore Fluid filled

pore

Pore space stiffness

By combining cases A and B (mineral and dry) we come up with the following equation:

porosity. and stiffness, space poredry

modulus,bulk mineral modulus,bulk rock dry

: where,11

min

min

K

KK

KKK

dry

dry

6

stiffness space pore saturated ~

and modulus,bulk rock saturated

: where,~11

min

min

fluid

fluid

sat

msat

KK

KKKK

K

KKK

By combining cases A and C (mineral and saturated) we come up with the following equation :

Deriving the Gassmann equation

We can combine these two fundamental equations to

arrive at the simplest form of the Gassmann equation:

)( minminmin fluid

f

dry

dry

sat

sat

KK

K

KK

K

KK

K

Gassmann also showed that shear modulus is independent

of fluid content.

AVO 1.0 used Gassmann’s equation for fluid substitution

and dry rock compressibility for porosity change, as well as:

.,saturationwater :where

,11

,,)3/4(

min

drysatw

whcwwsat

sat

S

sat

satP

S

)S(ρSρ)(ρρ

VK

V

8

Using the Gassmann equations

An example of using

Gassmann’s

equations, where P

and S-wave velocity

are shown as a

function of gas

saturation in the

reservoir.

Note that P-wave

velocity drops

dramatically as we

add gas, but that S-

wave velocity only

increases slightly.

Biot’s 1941 paper

Biot’s 1941 paper, “General Theory of Three-Dimensional

Consolidation”, pre-dated Gassmann’s work by 10 years.

Biot was an engineer, so he uses E (Young’s modulus)

and n (Poisson’s ratio) to derive his formula.

However, we can convert his final equations to the other

two elastic constant forms to get:

modulus. fluid the and t,coefficien Biot the

where,)2(

)1(

2

2

M

MKK

M

drysat

drysat

9

These two equivalent formulations show us that it is the

effect of the second fluid term, not the particular elastic

constants, that is important.

Relating Biot and Gassmann

To equate the Biot and Gassmann formulations, note that Gassmann’s equation can also be written:

2

minmin

2

min

1

1

K

K

KK

K

K

KKdry

fluid

dry

drysat

,1

and 1minmin KKMK

K

fluid

dry

MKK drysat2

If we let:

we see that Gassmann’s equation is identical to Biot’s:

10

The generalized LMR method

Combining the work of Biot and the work of Ken Hedlin (2000), Russell, Hedlin, Hilterman and Lines (Geophysics, 2003) proposed a generalization of the LMR equation:

)()()( 2222 drySdryP sfVVf

Note that: 3

42

2

2

drydry

dryS

Pdry

K

V

V

Murphy et al (1993) measured values of Kdry and for

clean quartz sandstones, and found a ratio of 0.9.

This was the value used by Ken Hedlin (2000).

Note that a value of 2 gives us the LMR approach.

A table of values is shown on the next page. 11

Table of values for the ratio

(Vp/Vs )dry2

Vp/Vs dry s dry K dry / dry/

4.000 2.000 0.333 2.667 2.000

3.333 1.826 0.286 2.000 1.333

3.000 1.732 0.250 1.667 1.000

2.500 1.581 0.167 1.167 0.500

2.333 1.528 0.125 1.000 0.333

2.250 1.500 0.100 0.917 0.250

2.233 1.494 0.095 0.900 0.233

2.000 1.414 0.000 0.667 0.000

1.333 1.155 -1.000 0.000 -0.667

(4)

(2)

(1)

In the above table (1) corresponds to K-, (2) to ,

(3) to the Murphy/Hedlin value, and (4) to a clean

unconsolidated sand. 12

(3)

13

P-wave and S-wave Inversion

As shown above, the practical implementation of this

method uses the inverted P and S impedances.

To find the optimum ratio, well logs should be used.

The next slides show an example from Whiterose,

courtesy of Ken Hedlin and Husky Oil.

We will use three values for 2dry: 1.333, 2, and 2.333.

AI = VP SI = VS

222 SIAIf dry

Gas sand

Oil sand

Wet sand

Cretaceous

Shale

Limestone

Vs Vp Den Porosity

85m

97m

95m

Courtesy, Ken Hedlin and Husky Oil 14

Whiterose well log example

f vs s with c = 1.333

rho*f vs rho*s for c = 1.333

0.00

0.50

1.00

1.50

2.00

2.50

3.00

3.50

4.00

4.50

5.00

0.00 2.00 4.00 6.00 8.00

rho*f

rho

*s

Shale Gas Oil Wet

0.00

1.00

2.00

3.00

4.00

5.00

6.00

7.00

0.00 1.00 2.00 3.00 4.00

rho

*s

rho*f

rho*f vs rho*s for c = 2

Shale Gas Oil Wet

1.00

2.00

3.00

4.00

5.00

6.00

7.00

8.00

0.00 1.00 2.00 3.00 4.00

rho

*s

rho*f

rho*f vs rho*s for c = 2.333

Shale Gas Oil Wet

15

AVO and poroelasticity

All three-term AVO equations can be written:

etc.impedance,impedance,,,,

as such parameters physical are ,, and

, / and of functions are and,,:where

321

22

SPVV

pandpp

VVcba

SP

SP

,)(2

2

2

2

1

1

p

pc

p

pb

p

paRPP

D

D

D

• For example, the original Aki-Richards equation uses

VP, VS and and the Fatti equation uses P-impedance,

S-impedance and .

16

AVO and the elastic constants

Gray et al. (1999) derived two new equations, one in

which the physical constants are , and , and the other for which they are K, and :

D

D

D

D

D

D

222

2

2

2

222

2

2

2

sec4

1

2

1sin2sec

3

11sec

3

1

4

1)(

sec4

1

2

1sin2sec

2

11sec

2

1

4

1)(

satsat

PP

satsat

PP

K

KR

R

These equations are also given on page 244 of the

textbook by Avseth et al.

Note the similarity of the two equations, which differ only

by the factors 1/2 and 1/3. 17

The generalized formulation

Russell, Gray and Hampson (Geophysics, 2011) re-

formulated the Aki-Richards equation using f, and :

where:

D

D

D cb

f

faRPP )(

dryS

Pdry

satS

Psat

satsat

dry

sat

dry

V

V

V

Vc

b

a

2

22

2

222

2

2

2

2

2

2

2

2

and ,sec4

1

2

1

sin2

sec4

sec44

1

18

Some observations

The following comments can be made about the general

formulation:

If we substitute dry2 = 2 into the previous formulation, we

obtain the Gray et al. (1999) expression for , , .

If we substitute dry2 = 4/3 into the previous formulation, we

obtain the Gray et al. (1999) expression for K, , .

Again, the optimum value should be determined from well

logs and will probably be in the order of 2.333 for clean

sands or nearer 2.8 for deeper sediments, as found by

Dillon et al. (TLE, 2003) for measurements in Brazil.

The next few slides show a case study of this method.

19

Real data study – Input gathers

We applied the f-- method to a Class 3 gas sand from

Alberta. The super-gathers are shown above, with the zone

of interest highlighted. Since the far angle is at 30o, the

density term extraction is considered unreliable. 20

Df/f vs D/ results

Here is a comparison

of the fluid result (top)

with the shear

modulus result

(bottom).

Note the change in

polarity at the gas

sand when comparing

the two results.

21

Df/f

D/

Pore space stiffness

Recall that in version 1.0 of the software we used the dry pore space stiffness equation to compute porosity effects:

porosity. and stiffness, space poredry

modulus,bulk mineral modulus,bulk rock dry

: where,11

K

KK

KKK

mdry

mdry

22

The use of this method was questioned by several of our clients (they preferred the critical porosity method) so I prepared a CREWES paper in 2007 that analyzed the two methods uses Han’s classic dataset.

The results are shown on the next slide.

Following this slide, we show the basics of the method for computed porosity change.

Best fits for constant pressure

K /Km = 0.162

RMSE = 0.039 c = 34.3%

RMSE = 0.058

This figure (from Russell and Smith, 2007) shows the fit of pore space

stiffness (left) and critical porosity (right) to a set of measured values at

constant pressure and differing porosity (Han, 1986). The pore space

stiffness method gives a smaller error than the critical porosity method.

Modeling Kdry versus porosity

Graphically, we model Kdry at a new porosity new using a

calibration porosity cal, moving along the K curve.

cal new

Knew

Kcal

Km

Constant K

curve

Note that

K reduces

to Km at 0%

porosity, as

it should.

24

Modeling Kdry versus porosity

Mathematically, this is done as follows:

mcaldrycal

new

mnewdry KKKK

1111

__

25

This work lead to another CREWES paper in 2011, where I

proposed a new approach to the Rock Physics Template, or

RPT.

I will first review the concepts of the RPT and then discuss

this new approach.

I will finish with both a log and real data example.

The rock physics template (RPT)

Ødegaard and Avseth

(2003) proposed a

technique they called

the rock physics

template (RPT), in

which the fluid and

mineralogical content of

a reservoir could be

estimated on a crossplot

of Vp/Vs ratio against

acoustic impedance, as

shown here.

from Ødegaard and Avseth (2003) 26

The Ødegaard/Avseth RPT

Ødegaard and Avseth (2003) compute Kdry and dry as a

function of porosity using Hertz-Mindlin (HM) contact

theory and the lower Hashin-Shtrikman bound:

member.-endporosity high and ratio, sPoisson' mineral grain,per

contacts ,modulusshear andbulk mineral ,,pressure confining

,)1(2

)1(3

)2(5

44 ,

)1(18

)1(

,2

89

6 where,

3

4/1/

3

4

)3/4(

/1

)3/4(

/

3

1

22

2223

1

22

222

1

1

cm

mm

m

mc

m

mHM

m

mcHM

HMHM

HMHMHMHM

m

c

HM

cdry

HM

HMm

c

HMHM

cdry

nKP

Pn

Pn

K

K

Kz

zz

KKK

n

n

n

n

n

They then use standard Gassmann theory for the fluid

replacement process. 27

The new Gassmann RPT

mcaldrycal

new

mnewdry

1111

__

I proposed a new approach to the rock physics template, in

which we still use Gassmann for saturation change but use

pore space stiffness to compute the porosity change.

The new approach uses the same formulation for shear

modulus as for the bulk modulus (Qing Li suggested this):

28

The next slide shows a comparison of the two methods in

terms of the final dry modulus ratio.

We will then look at a case study on both logs and real

data.

Comparison of the methods for the modulus ratio

A comparison between the two methods and the constant ratio empirical

result. The plot on the left shows the dry rock K/ ratio as a function of porosity (0 to 40%) and the plot on the right shows the dry shear

modulus as a function of porosity for only the first 10% of porosity:

The new approach is closer to the experimental results of Murphy et al.,

except near 0% porosity, where it correctly predicts the mineral value. 29

Vp/Vs vs P-impedance from logs

The Vp and logs were measured and Vs was computed using

the mud-rock line in the shales and wet sands and the

Gassmann equations in the gas sands.

Now we will

compare our

templates to

real data. This

plot shows well

log data from a

gas sand in the

Colony area of

Alberta.

P-impedance (m/s*g/cc)

Vp/V

s r

atio

Gas sands

Brine sands

Shales

Cemented sands

4500 11000

1.5

3.0

30

Vp/Vs vs P-impedance from inversion

The results of a

simultaneous

pre-stack

inversion from

the same area.

Note that the

range of values

is less extreme

than on the log

data due to the

bandlimited

nature of the

seismic data.

Gas sands

Brine sands

Shales

Cemented sands

P-impedance (m/s*g/cc)

Next, we show the log and seismic data superimposed on the

RPTs, where the log data has been integrated to time.

Vp/V

s r

atio

5200

1.8

2.9

6800

31

Avseth/Ødegaard Rock Physics Template

30% Porosity 20% Porosity Seismic (Vp/Vs shifted) Log data 32

Pressure

Clay content

Shale

Porosity

Cement

Gas

Gas Sands

New Rock Physics Template

30% Porosity 20% Porosity Seismic (Vp/Vs shifted) Log data 33

Pressure

Clay content

Shale

Porosity

Cement

Gas

Gas Sands

Anisotropic Hooke’s Law

Here is the stiffness form of Hooke’s law, which relates the

stress tensor to the strain tensor, for the orthorhombic

anisotropic case, requiring 9 independent stiffness terms:

ts.coefficien stiffness strain, stress,:where

,

00000

00000

00000

000

000

000

12

13

23

33

22

11

66

55

44

332313

232212

131211

12

13

23

33

22

11

ijijij c

c

c

c

ccc

ccc

ccc

s

s

s

s

s

s

s

Note that the isotropic case, which we have been

discussing so far, requires only 2 independent terms.

Anisotropic Biot-Gassmann

The anisotropic Gassmann equations can be written in the

following Biot-type form (Gurevich, 2003):

Mcc jidry

ij

sat

ij

Note that the anisotropic Biot coefficients are simply sums

over the first three columns of the stiffness matrix.

I will next apply these equations to an orthorhombic

sandstone example from Dillen (2000).

3

1

3

1

*

minmin

*

min

min

3

1

9

1,

)/(1)/(1

,6,5,4,0,3,2,1,3

1 :where

i j

dry

ij

fluid

mn

dry

mn

m

cKKKKK

KM

mmK

c

Orthorhombic Sandstone Example

Conclusions

In this talk, I outlined my long “relationship” with Biot and

Gassmann.

It started in 1985, with version 1.0 of our AVO program.

I then used their theories, along with ideas from Ken Hedlin

and Fred Hilterman, to generalize the LMR process.

With Dave Gray, we used their theories to generalize two

separate linearized AVO expressions.

Most recently, I used the pore space compressibility concept

to build a new rock physics template.

This last work validates the assumptions that we made in

version 1.0 of AVO about computing porosity change.

My current interest involves using Biot and Gassmann in the

anisotropic world.

37

References

Biot, M. A., 1941, General theory of three-dimensional consolidation,

Journal of Applied Physics, 12, 155-164.

Dillon, L., Schwedersky, G., Vasquez, G., Velloso, R. and Nunes, C., 2003,

A multiscale DHI elastic attributes evaluation: The Leading Edge, 22, no.

10, 1024-1029.

Gassmann, F., 1951, Uber die Elastizitat poroser Medien: Vierteljahrsschrift

der Naturforschenden Gesellschaft in Zurich, 96, 1-23.

Goodway, W., Chen, T., and Downton, J., 1997, Improved AVO fluid

detection and lithology discrimination using Lame petrophysical parameters:

67th Ann. Internat. Mtg., Soc. Expl. Geophys., Expanded Abstracts, 183–

186.

Gray, F., Chen, T. and Goodway, W., 1999, Bridging the gap: Using AVO to

detect changes in fundamental elastic constants, 69th Ann. Int. Mtg: SEG,

852-855.

38

References

Han, D., 1986, Effects of porosity and clay content on acoustic properties

of sandstones and unconsolidated sediments: Ph.D. dissertation,

Stanford.

Hedlin, K., 2000, Pore space modulus and extraction using AVO: 70th

Ann. Internat. Mtg., Soc. Expl. Geophys., Expanded Abstracts, 170-173.

Mavko, G., and T. Mukerji, 1995, Seismic pore space compressibility and

Gassmann's relation: Geophysics, 60, 1743-1749.

Murphy, W., Reischer, A., and Hsu, K., 1993, Modulus Decomposition of

Compressional and Shear Velocities in Sand Bodies: Geophysics, 58,

227-239.

Nur, A., 1992, Critical porosity and the seismic velocities in rocks: EOS,

Transactions American Geophysical Union, 73, 43-66.

Ødegaard, E. and Avseth, P., 2003, Interpretation of elastic inversion

results using rock physics templates: EAGE, Expanded Abstracts.

39

References

Russell, B., Hedlin, K., Hilterman, F. and Lines, L., 2003, Fluid-property

discrimination with AVO: A Biot-Gassmann perspective: Geophysics,

68, 29-39.

Russell, B. H. and Smith, T., 2007, The relationship between dry rock

bulk modulus and porosity – An empirical study: CREWES Report,

Volume 19.

Russell, B.H. and Lines, L., 2011, A Gassmann consistent rock physics

template: CREWES Report, Volume 23.

Russell, B.H., Gray, D., and Hampson, D.P., 2011, Linearized AVO and

poroelasticity, Geophysics, 76, no. 3, C19-C29.

40