Reservoir Geophysics : Brian Russell Lecture 1

AVO and Inversion - Part 1

Introduction and Rock Physics

Dr. Brian Russell

Overview of AVO and Inversion

This tutorial is a brief introduction to the Amplitude

Variations with Offset, or Amplitude Versus Offset

(AVO), and pre-stack inversion methods.

I will briefly review how the interpretation of seismic

data has changed through the years.

I will then look at why AVO and pre-stack inversion

was an important step forward for the interpretation

of hydrocarbon anomalies.

Finally, I will show why the AVO and pre-stack

inversion responses are closely linked to the rock

physics of the reservoir.

2

A Seismic Section

The figure above shows a stacked seismic section recorded over the shallow

Cretaceous in Alberta. How would you interpret this section?

3

Structural Interpretation

Your eye may first go to an anticlinal seismic event between 630 and 640 ms. Here, it

has been picked and called H1. A seismic interpreter prior to 1970 would have looked

only at structure and perhaps have located a well at CDP 330.

4

Gas Well Location

And, in this case, he or she would have been right! A successful gas well was drilled

at that location. The figure above shows the sonic log, integrated to time, spliced on

the section. The gas sand top and base are shown as black lines on the log.

5

“Bright Spots”

But this would have been a lucky guess, since structure alone does not tell you that a

gas sand is present. A geophysicist in the 1970’s would have based the well on the

fact that there is a “bright spot” visible on the seismic section, as indicated above.

6

What is a “Bright Spot”?

To understand “bright spots”, recall the definition of the zero-offset reflection coefficient, shown in the figure above. R0 , the reflection coefficient, is the amplitude of the seismic trough shown. Note also that the product of density, r, and P-wave velocity, V, is called acoustic impedance.

1122

11220

VV

VVR

rr

rr

Seismic

raypath

Interface at

depth = d

r1 V1

r2 V2

Reflection at time

t = 2d/V1

Geology SeismicSurface

Seismic

Wavelet

Shale

Gas Sand

7

This figure, from

Gardner et al. (1974),

shows a big difference

between shale and gas

sand velocity at

shallow depths in the

Gulf of Mexico. The

paper also derived the

“Gardner” equation,

which states that

density and velocity are

related by the equation

r = 0.23 V 0.25

Thus, we would expect

a large reflection

coefficient, or “bright

spot”, for shallow gas

sands.

Difference between shale and gas

sand velocity at shallow depth.

Gardner’s results for GOM

8

The AVO Method

“Bright spots” can

be caused by

lithologic variations

as well as gas

sands.

Geophysicists in

the 1980’s looked at

pre-stack seismic

data and found that

amplitude change

with offset could be

used to explain gas

sands (Ostrander,

1984). This example

is a Class 3 gas

sand, which we will

discuss later.

9

What causes the AVO Effect?

The traces in a seismic gather reflect from the subsurface at increasing

angles of incidence q. The first order approximation to the reflection

coefficients as a function of angle is given by adding a second term to the

zero-offset reflection coefficient:qq 2

0 sin)( BRR

q1q2q3

Surface

Reflectorr1 VP1 VS1

r2 VP2 VS2

B is a gradient term which produces the AVO effect. It is dependent on

changes in density, r, P-wave velocity, VP, and S-wave velocity, VS.

10

This diagram shows a schematic diagram of (a) P, or compressional, waves,

(b) SH, or horizontal shear-waves, and (c) SV, or vertical shear-waves, where

the S-waves have been generated using a shear wave source (Ensley, 1984).

(a) (b) (c)

P and S-Waves

11

Note that we can also record S wave information.

Why is S-wave Velocity Important?

12

The plot on the left

shows P and S-wave

velocity plot as a

function of gas

saturation (100% gas

saturation = 0% Water

Saturation), computed

with the Biot-

Gassmann equations.

Note that P-wave

velocity drops

dramatically, but S-

wave velocity only

increases slightly

(why?). This will be

discussed in the next

section.

AVO Modeling

Based on AVO theory and the rock physics of the reservoir, we can perform AVO

modeling, as shown above. Note that the model result is a fairly good match to the

offset stack. Poisson’s ratio is a function of Vp/Vs ratio and will be discussed in the

next chapter.

P-wave Density S-wavePoisson’s

ratioSynthetic Offset Stack

13

AVO Attributes

Intercept: A

Gradient: B

AVO Attributes are

used to analyze

large volumes of

seismic data,

looking for

hydrocarbon

anomalies.

14

Cross-Plotting of Attributes

One of the AVO methods that we will be

discussing later in the course involves

cross-plotting the zero-offset reflection

coefficient (R0, usually called A), versus the

gradient (B), as shown on the left.

As seen in the figure below, the highlighted

zones correspond to the top of gas sand

(pink), base of gas sand (yellow), and a hard

streak below the gas sand (blue).

Gradient (B)

Intercept (A)

15

AVO Inversion

A new tool combines

inversion with AVO

Analysis to enhance the

reservoir discrimination.

Here, we have inverted for

P-impedance and Vp/Vs

ratio, cross-plotted and

identified a gas sand.

16

Gas

Sand

Summary of AVO Methodology

17

Input NMO-corrected Gathers

Recon Methods InversionModeling

Intercept

Gradient

Partial

Stacks

Zoeppritz

Synthetics

Wave Eq.

Synthetics

Cross

PlotLMR

Elastic

ImpedanceSimultaneous

Inversion

Perform optimum processing sequence

Rock Physics

Modeling

Conclusions

Seismic interpretation has evolved over the years,

from strictly structural interpretation, through “bright

spot” identification, to direct hydrocarbon detection

using AVO and pre-stack inversion.

In this short course I will elaborate on the ideas that

have been presented in this short introduction.

As a starting point, the next section I will discuss the

principles of rock physics in more detail.

I will then move to AVO modeling and analysis.

Finally, I will look at AVO and pre-stack inversion

analysis on real seismic data.

18

Rock Physics and Fluid Replacement Modeling

Pores / FluidRock Matrix

The AVO response is dependent on the properties of P-wave velocity (VP),

S-wave velocity (VS), and density (r) in a porous reservoir rock. As shown

below, this involves the matrix material, the porosity, and the fluids filling

the pores:

Basic Rock Physics

20

)1()1( whcwwmsat SρSρρρ

.subscriptswatern,hydrocarbo

matrix,saturated,,

,saturationwater

porosity,

density,:where

wsat,m,hc

wS

ρ

This is illustrated in the next graph.

Density effects can be modeled with the following equation:

Density

21

Density versus Water Saturation

Density vs Water Saturation

Sandstone with Porosity = 33%

Densities (g/cc): Matrix = 2.65, Water = 1.0,

Oil = 0.8, Gas = 0.001

1.6

1.7

1.8

1.9

2

2.1

2.2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Water Saturation

Den

sit

y

Oil Gas

Here is a plot of density

vs water saturation for a

porous sand with the

parameters shown,

where we have filled the

pores with either oil or

gas.

In the section on AVO

we will model both the

wet sand and the 50%

saturated gas sand.

Note that these density

values can be read off

the plot and are:

rwet = 2.11 g/cc

rgas = 1.95 g/cc

22

P and S-Wave Velocities

Unlike density, seismic velocity involves the deformation of a rock as a

function of time. As shown below, a cube of rock can be compressed, which

changes its volume and shape or sheared, which changes its shape but not

its volume.

23

P-waves S-waves

The leads to two different types of velocities:

P-wave, or compressional wave velocity, in which the direction of particle motion is in the same direction as the wave movement.

S-wave, or shear wave velocity, in which the direction of particle motion is at right angles to the wave movement.

P and S-Wave Velocities

24

r

2PV

r

SV

where: = the first Lamé constant,

= the second Lamé constant,

and r = density.

The simplest forms of the P and S-wave velocities are derived for

non-porous, isotropic rocks. Here are the equations for velocity

written using the Lamé coefficients:

Velocity Equations using and

25

r

3

4

K

VP r

SV

where: K = the bulk modulus, or the reciprocal of compressibility.

= + 2/3

= the shear modulus, or the second Lamé constant,

and r = density.

Another common way of writing the velocity equations is with

bulk and shear modulus:

Velocity Equations using K and

26

Poisson’s Ratio from strains

The Poisson’s ratio, , is defined as the negative of the ratio

between the transverse and longitudinal strains:

If we apply a compressional

force to a cylindrical piece of

rock, as shown on the right, we

change its shape.

)//()/( LLRR

R

R+R

L+L L

F (Force)

F

The longitudindal strain is given

by L/L and the transverse strain

is given by R/R.

(In the typical case shown above, L is negative, so is positive)27

22

22

2

S

P

V

V:where

This formula is more useful in our calculations than the formula given

by the ratio of the strains. The inverse to the above formula, allowing

us to derive VP or VS from , is given by:

12

222

A second way of looking at Poisson’s ratio is to use the ratio of VP to VS,

and this definition is given by:

Poisson’s Ratio from velocity

28

Vp/Vs vs Poisson's Ratio

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0 1 2 3 4 5 6 7 8 9 10

Vp/Vs

Po

isso

n's

Rati

o

Gas Case Wet Case

Poisson’s Ratio vs VP/VS ratio

29

If VP/VS = 2, then = 0

If VP/VS = 1.5, then = 0.1 (Gas Case)

If VP/VS = 2, then = 1/3 (Wet Case)

If VP/VS = , then = 0.5 (VS = 0)

Poisson’s Ratio

From the previous figure, note that there are several values of

Poisson’s ratio and VP/VS ratio that are important to remember.

30

Note also from the previous figure that Poisson’s ratio can

theoretically be negative, but this has only been observed for

materials created in the lab (e.g. Goretex and polymer foams).

A plot of velocity versus

water saturation using

the above equation. We

used a porous sand with

the parameters shown

and have filled the pores

with either oil or gas.

This equation does not

hold for gas sands, and

this lead to the

development of the Biot-

Gassmann equations.

Velocity vs Water Saturation

Wyllie's Equation

Porosity = 33%

Vmatrix = 5700 m/s, Vw = 1600 m/s,

Voil = 1300 m/s, Vgas = 300 m/s.

500

1000

1500

2000

2500

3000

3500

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Water Saturation

Velo

cit

y (

m/s

ec)

Oil Gas

Velocity in Porous Rocks

Velocity effects can be modeled by the volume average equation:

V/t,)S(tSt)(tt whcwwmsat 1 where11

31

sat

satsat

satP

KV

r

3

4

_

sat

satsatSV

r

_

Note that rsat is found using the volume average equation:

The volume average equation gives incorrect results for gas sands.

Independently, Biot (1941) and Gassmann (1951), developed a more

correct theory of wave propagation in fluid saturated rocks, especially gas

sands, by deriving expressions for the saturated bulk and shear moduli

and substituting into the regular equations for P and S-wave velocity:

The Biot-Gassmann Equations

)1()1( whcwwmsat SρSρρρ

32

drysat

In the Biot-Gassmann equations, the shear modulus does not change for

varying saturation at constant porosity. In equations:

The Biot-Gassmann Equations

To understand the Biot-Gassmann equations, let us update the figure we saw earlier to include the concepts of the “saturated rock” (which includes the in-situ fluid) and the “dry rock” (in which the fluid has been drained.)

Rock Matrix Pores and fluid

Dry rock

frame, or

skeleton

(pores

empty)

Saturated

Rock

(pores full)

33

2

2

1

1

m

dry

mfl

m

dry

drysat

K

K

KK

K

K

KK

Mavko et al, in The Rock Physics Handbook, re-arranged the above

equation to give a more intuitive form:

)( flm

fl

drym

dry

satm

sat

KK

K

KK

K

KK

K

where sat = saturated rock, dry = dry frame, m = mineral, fl = fluid,

and = porosity.

(1)

(2)

The Biot-Gassmann bulk modulus equation is as follows:

Biot-Gassmann – Saturated Bulk Modulus

34

Biot’s Formulation

Biot defines b (the Biot coefficient) and M (the fluid modulus) as:

,1

and ,1mflm

dry

KKMK

K bb

Equation (1) then can be written as: MKK drysat

2b

If b = 0 (or Kdry = Km) this equation simplifies to: drysat KK

If b = 1 (or Kdry= 0), this equation simplifies to:

mflsat KKK

11

Physically, b = 0 implies we have a non-porous rock, and b = 1 implies we

have particles in suspension (and the formula given is called Wood’s

formula). These are the two end members of a porous rock.

35

Ksandstone = 40 GPa,

Klimestone = 60 GPa.

We will now look at how to get estimates of the various bulk modulus

terms in the Biot-Gassmann equations, starting with the bulk modulus of

the solid rock matrix. Values will be given in gigaPascals (GPa), which

are equivalent to 1010 dynes/cm2.

The bulk modulus of the solid rock matrix, Km is usually taken from

published data that involved measurements on drill core samples.

Typical values are:

The Rock Matrix Bulk Modulus

36

hc

w

w

w

fl K

S

K

S

K

11

Equations for estimating the values of brine, gas, and oil bulk modulii are

given in Batzle and Wang, 1992, Seismic Properties of Pore Fluids,

Geophysics, 57, 1396-1408. Typical values are:

Kgas = 0.021 GPa, Koil = 0.79 GPa, Kw = 2.38 GPa

fl

w

hc

where the bulk modulus of the fluid,

the bulk modulus of the water,

and the bulk modulus of the hydrocarbon.

K

K

K

The fluid bulk modulus can be modeled using the following equation:

The Fluid Bulk Modulus

37

The key step in FRM is calculating a value of Kdry. This can be done in several ways:

(1) For known VS and VP, Kdry can be calculated by first calculating Ksat

and then using Mavko’s equation (equation (2)), given earlier.

(2) For known VP, but unknown VS, Kdry can be estimated by:

(a) Assuming a known dry rock Poisson’s ratio dry. Equation (1) can

then be rewritten as a quadratic equation in which we solve for Kdry.

(b) Using the Greenberg-Castagna method, described later.

Estimating Kdry

38

In the next few slides, we will look at the computed responses for

both a gas-saturated sand and an oil-saturated sand using the

Biot-Gassmann equation.

We will look at the effect of saturation on both velocity (VP and VS)

and Poisson’s Ratio.

Keep in mind that this model assumes that the gas is uniformly

distributed in the fluid. Patchy saturation provides a different

function. (See Mavko et al: The Rock Physics Handbook.)

Data Examples

39

Velocity vs Saturation of Gas

Velocity vs Water Saturation - Gas Case

Sandstone with Phi = 33%, Density as previous figure for gas,

Kmatrix = 40 Gpa, Kdry = 3.25 GPa, Kw = 2.38 Gpa,

Kgas = 0.021 Gpa, Shear Modulus = 3.3. Gpa.

1000

1200

1400

1600

1800

2000

2200

2400

2600

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Sw

Velo

cit

y (

m/s

)

Vp Vs

A plot of velocity vs water

saturation for a porous gas

sand using the Biot-Gassmann

equations with the parameters

shown.

In the section on AVO we will

model both the wet sand and

the 50% saturated gas sand.

Note that the velocity values

can be read off the plot and

are:

VPwet = 2500 m/s

VPgas = 2000 m/s

VSwet = 1250 m/s

VSgas = 1305 m/s

40

Poisson’s Ratio vs Saturation of Gas

Poisson's Ratio vs Water Saturation - Gas Case

Sandstone with Phi = 33%, Density as previous figure for gas,

Kmatrix = 40 Gpa, Kdry = 3.25 GPa, Kw = 2.38 Gpa,

Kgas = 0.021 Gpa, Shear Modulus = 3.3. Gpa.

0

0.1

0.2

0.3

0.4

0.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Sw

Po

isso

n's

Rati

o

A plot of Poisson’s ratio vs

water saturation for a porous

gas sand using the Biot-

Gassmann equations with the

parameters shown.

In the section on AVO we will

model both the wet sand and

the 50% saturated gas sand.

Note that the Poisson’s ratio

values can be read off the plot

and are:

wet = 0.33

gas = 0.12

41

Velocity vs Saturation of Oil

Velocity vs Water Saturation - Oil Case

Sandstone with Phi = 33%, Density as previous figure for oil,

Kmatrix = 40 Gpa, Kdry = 3.25 GPa, Kw = 2.38 Gpa,

Koil = 1.0 Gpa, Shear Modulus = 3.3. Gpa.

1000

1200

1400

1600

1800

2000

2200

2400

2600

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Sw

Velo

cit

y (

m/s

)

Vp Vs

A plot of velocity vs water

saturation for a porous oil

sand using the Biot-

Gassmann equations with

the parameters shown.

Note that there is not much

of a velocity change.

However, this is for “dead”

oil, with no dissolved gas

bubbles, and most oil

reservoirs have some

percentage of dissolved

gas.

42

Poisson’s Ratio vs Saturation of Oil

Poisson's Ratio vs Water Saturation - Oil Case

Sandstone with Phi = 33%, Density as previous figure for oil,

Kmatrix = 40 Gpa, Kdry = 3.25 GPa, Kw = 2.38 Gpa,

Koil = 1.0 Gpa, Shear Modulus = 3.3. Gpa.

0

0.1

0.2

0.3

0.4

0.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Sw

Po

isso

n's

Rati

o

A plot of Poisson’s ratio vs

water saturation for a porous

oil sand using the Biot-

Gassmann equations with the

parameters shown.

Note that there is not much of

a Poisson’s ratio change.

However, again this is for

“dead” oil, with no dissolved

gas bubbles, and most oil

reservoirs have some

percentage of dissolved gas.

43

Fluid substitution in carbonates

In general carbonates are thought to have a smaller fluid sensitivity than

clastics. This is a consequence of the fact that they are typically stiffer (i.e.

have larger values of Km and Kdry ) implying a smaller Biot coefficient b and

hence fluid response.

This general observation is complicated by the fact that carbonates often

contain irregular pore shapes and geometries.

High aspect ratio pores make the rock more compliant and thus more

sensitive to fluid changes.

Aligned cracks require the use of the anisotropic Gassmann equation,

resulting in the saturated bulk modulus being directionally dependent.

Gassmann assumed that pore pressure remains constant during wave

propagation. If the geometry of the pores and cracks restrict the fluid

flow at seismic frequencies then the rock will appear stiffer.

All these factors make the application of the Biot-Gassmann fluid

substitution in carbonates more complex.

44

Kuster-Toksöz model

The Kuster-Toksöz model allows to estimate properties of the rocks with ellipsoidal pores, filled up with any kind of fluid.

• The Kuster-Toksöz model was developed in 1974

• Based on ellipsoidal pore shape (Eshelby, 1957)

• Pore space described as a collection of pores of

different aspect ratios

a

b

Aspect Ratio α= b/aCourtesy of A. Cheng(2009)

In the appendix, we show how to compute the Kuster-Toksözmodel values Tiijj and F.

Kuster-Toksöz model

Pores in the rock according to Kuster-Toksöz model.

Courtesy of A. Cheng(2009)

NO

RM

AL

IZE

D V

EL

OC

ITY

(V

/V M

AT

RIX

)

1.0

0.95

0.9

0.85

0.8

a = 1.0

0.1

0.05

0.01WATER-SATURATED

GAS-SATURATED

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

P Wave

S Wave

POROSITY (%)

Kuster-Toksöz model

Pore shape (aspect ratio a) effect on velocities.

Toksöz et al., (1976)

48

The Keys-Xu method

Keys and Xu (2002) give a method for computing the dry

rock moduli as a function of porosity, mineral moduli and

pore aspect ratio.

The equations are as follows, where p and q are functions

of the scalars given by Kuster and Toksöz (1974):

mineral. of ratioaspect and clay, of ratioaspect

before, as ,1

1,

1

),(5

1 ,)(

3

1

where,)1( and )1(

21

21

2

1

2

1

aa

aa

clayclay

k

kk

k

kiijjk

q

m

p

mdry

Vf

Vf

FfqTfp

KK

49

The Keys-Xu method

Here is a plot of the

results of the Keys

and Xu (2002)

method for the dry

rock bulk modulus:

When multiple pore fluids are present, Kfl is usually calculated by a Reuss

averaging technique (see Appendix 2):

Kfl vs Sw and Sg

0

0.5

1

1.5

2

2.5

3

0 0.25 0.5 0.75 1

Water saturation (fraction)

Bu

lk m

od

ulu

s (G

pa

)This averaging

technique assumes

uniform fluid

distribution!

-Gas and liquid must

be evenly distributed

in every pore.

This method heavily biases compressibility of the combined fluid to

the most compressible phase.

g

g

o

o

w

w

fl K

S

K

S

K

S

K

1

Patchy Saturation

50

When patch sizes are large with respect to the seismic wavelength, Voigt

averaging (see Appendix 2) gives the best estimate of Kfl (Domenico, 1976):

When patch sizes are of intermediate size, Gassmann substitution should

be performed for each patch area and a volume average should be made.

This can be approximated by using a power-law averaging technique,

which we will not discuss here.

ggoowwfl KSKSKSK

When fluids are not uniformly mixed, effective modulus values cannot be

estimated from Reuss averaging. Uniform averaging of fluids does not

apply.

Patchy Saturation

51

Gassmann predicted velocities

Unconsolidated sand matrix

Porosity = 30%

100% Gas to 100% Brine saturation

1.5

1.7

1.9

2.1

2.3

2.5

0 0.25 0.5 0.75 1

Water Saturation (fraction)

Vp

(k

m/s

)

Patchy

Voigt

Reuss

Patchy Saturation

52

SP VV12

22

This will be illustrated in the next few slides.

Note that for a constant Poisson’s ratio, the intercept is zero:

smVV SP /136016.1

The mudrock line is a linear relationship between VP and VS

derived by Castagna et al (1985):

The Mudrock Line

53

ARCO’s original mudrock derivation

(Castagna et al, Geophysics, 1985)

The Mudrock Line

54

0

2000

2000

4000

6000

1000 3000 40000

1000

3000

5000

VP (m/s)

VS(m/s)

Mudrock Line

Gas Sand

The Mudrock Line

55

0

2000

2000

4000

6000

1000 3000 40000

1000

3000

5000

VP (m/s)

VS(m/s)

Mudrock Line

Gas Sand

= 1/3

or

VP/VS = 2

The Mudrock Line

56

VP

(m/s)

0

2000

2000

4000

6000

1000 3000 40000

1000

3000

5000

VS(m/s)

Mudrock Line

Gas Sand

= 1/3 or

VP/VS = 2

= 0.1 or

VP/VS = 1.5

The Mudrock Line

57

Using the regression coefficients given above, Greenberg and Castagna

(1992) first propose that the shear-wave velocity for a brine-saturated rock

with mixed mineral components can be given as a Voigt-Reuss-Hill

average of the volume components of each mineral.

PS

PS

PPS

PS

VskmV

VskmV

VVskmV

VskmV

770.0/867.0 :Shale

583.0/078.0 :Dolomite

055.0017.1/031.1 :Limestone

804.0/856.0 :Sandstone

2

Greenberg and Castagna (1992) extended the previous mud-rock

line to different mineralogies as follows, where we have now

inverted the equation for VS as a function of VP:

The Greenberg-Castagna method

58

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) 59

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.

Hertz-Mindlin contact theory assumes that the porous rock

can be modeled as a packing of identical spheres, and the

effective bulk and shear moduli are computed from:

member.-endporosity high and

ratio, s Poisson'mineral grain,per contacts

,modulusshear mineral ,pressure confining :where

,)1(2

)1(3

)2(5

44 ,

)1(18

)1( 3

1

22

2223

1

22

222

c

m

m

m

mc

m

meff

m

mceff

n

P

Pn

Pn

K

The rock physics template (RPT)

60

The lower Hashin-Shtrikman bound is then used to compute

the dry rock bulk and shear moduli as a function of porosity

with the following equations:

modulus.bulk mineral and 2

89

6

:where,3

4/1/

3

4

)3/4(

/1

)3/4(

/

1

1

m

effeff

effeffeff

m

c

eff

cdry

eff

effm

c

effeff

cdry

KK

Kz

zzz

KKK

Standard Gassmann theory is then used for the fluid

replacement process.

The rock physics template (RPT)

61

Here is the RPT for a range of porosities and water saturations, in a

clean sand case. We will build this template in the next exercise.

The rock physics template (RPT)

62

An understanding of rock physics is crucial for the

interpretation of AVO anomalies.

The volume average equation can be used to model

density in a water sand, but this equation does not

match observations for velocities in a gas sand.

The Biot-Gassmann equations match observations well

for unconsolidated gas sands.

When dealing with more complex porous media with

patchy saturation, or fracture type porosity (e.g.

carbonates), the Biot-Gassmann equations do not hold,

and we move to the Kuster-Toksöz approach.

The ARCO mudrock line is a good empirical tool for the

wet sands and shales.

Conclusions

63

,(34

1

,)1()1(

)2(2

1

,)2()43)(3(2

)43()53(2

)(2

311

,3

4

2

5

2

3)(

2

31:where

,12

)( and ,3

)(

4

2

2

3

2

2

1

42

987654

432

1

fgRgfA

F

RgfRA

F

ffgRfgRBAA

RBfgR

fgAF

fgRfgAF

FF

FFFFFF

FFF

F

FTiijj

a

a

aa

64

Appendix: The Kuster-Toksöz values

ratio.aspect pore and )23(1

)1(cos)1(

,43

3

,3

,1 ),43()1(

),43)(1(352

)1(2

21

),43(35394

2

),43)(1(11

),43(3

4

2

2

2/121

2/32

9

8

7

6

5

aa

a

aaaa

a

fg

fK

R

K

KBARBfRfRgAF

RfBRf

Rg

RAF

RBfgfRgfA

F

RfBfgRgAF

RBfgfgRAF

mm

m

m

f

65

Appendix: The Kuster-Toksöz values