This document provides an introduction to AVO and pre-stack inversion methods. It begins with a brief history of seismic interpretation, from purely structural interpretation to identifying "bright spots" to direct hydrocarbon detection using AVO and pre-stack inversion. It then discusses how AVO response is closely linked to rock physics properties like P-wave velocity, S-wave velocity, and density. The key concepts of AVO modeling and attributes are introduced. Finally, it provides an overview of rock physics and fluid replacement modeling using equations like Biot-Gassmann to model velocity and density changes with fluid saturation.

1. AVO and Inversion - Part 1
Introduction and Rock Physics
Dr. Brian Russell

2. 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

3. 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

4. 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

5. 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

6. “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

7. 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
1122
0
VV
VV
R
rr
rr



Seismic
raypath
Interface at
depth = d
r1 V1
r2 V2
Reflection at time
t = 2d/V1
Geology Seismic
Surface
Seismic
Wavelet
Shale
Gas Sand
7

8. 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

9. 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

10. 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

11. 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.

12. 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.

13. 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-wave
Poisson’s
ratio
Synthetic Offset Stack
13

14. AVO Attributes
Intercept: A
Gradient: B
AVO Attributes are
used to analyze
large volumes of
seismic data,
looking for
hydrocarbon
anomalies.
14

15. 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

16. 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

17. Summary of AVO Methodology
17
Input NMO-corrected Gathers
Recon Methods InversionModeling
Intercept
Gradient
Partial
Stacks
Zoeppritz
Synthetics
Wave Eq.
Synthetics
Cross
Plot
LMR
Elastic
Impedance
Simultaneous
Inversion
Perform optimum processing sequence
Rock Physics
Modeling

18. 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

19. Rock Physics and Fluid
Replacement Modeling

20. 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

21.  )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

22. 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
Density
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

23. 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

24. 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

25. r
 2
PV
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

26. 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

27. 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

28. 22
2
2
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

29. 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
Poisson'sRatio
Gas Case Wet Case
Poisson’s Ratio vs VP/VS ratio
29

30. 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).

31. 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
Velocity(m/sec)
Oil Gas
Velocity in Porous Rocks
Velocity effects can be modeled by the volume average equation:
V/t,)S(tSt)(tt whcwwmsat 1where11  
31

32. sat
satsat
satP
K
V
r

3
4
_

 sat
sat
satSV
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:

33. 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

34. 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

35. Biot’s Formulation
Biot defines b (the Biot coefficient) and M (the fluid modulus) as:
,
1
and,1
mflm
dry
KKMK
K b
b


Equation (1) then can be written as: MKK drysat
2
b
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

36. 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

37. 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

38. 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

39.  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

40. 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
Velocity(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

41. 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
Poisson'sRatio
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

42. 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
Velocity(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

43. 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
Poisson'sRatio
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

44. 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

45. 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/a
Courtesy of A. Cheng(2009)
In the appendix, we show how to compute the Kuster-Toksöz
model values Tiijj and F.

46. Kuster-Toksöz model
Pores in the rock according to Kuster-Toksöz model.
Courtesy of A. Cheng(2009)

47. NORMALIZEDVELOCITY(V/VMATRIX)
1.0
0.95
0.9
0.85
0.8
a = 1.0
0.1
0.05
0.01
WATER-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. 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.ofratioaspectandclay,ofratioaspect
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
V
f
V
f
FfqTfp
KK

49. 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:

50. 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)
Bulkmodulus(Gpa)
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

51. 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

52. 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(km/s)
Patchy
Voigt
Reuss
Patchy Saturation
52

53. SP VV
12
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

54. ARCO’s original mudrock derivation
(Castagna et al, Geophysics, 1985)
The Mudrock Line
54

55. 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

56. 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

57. 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

58. 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

59. 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

60.  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.-endporosityhighand
ratio,sPoisson'mineralgrain,percontacts
,modulusshearmineral,pressureconfining:where
,
)1(2
)1(3
)2(5
44
,
)1(18
)1( 3
1
22
2223
1
22
222






















c
m
m
m
mc
m
m
eff
m
mc
eff
n
P
P
n
P
n
K










The rock physics template (RPT)
60

61.  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.bulkmineraland
2
89
6
:where,
3
4/1/
3
4
)3/4(
/1
)3/4(
/
1
1










































m
effeff
effeffeff
m
c
eff
c
dry
eff
effm
c
effeff
c
dry
K
K
K
z
z
zz
KK
K












 Standard Gassmann theory is then used for the fluid
replacement process.
The rock physics template (RPT)
61

62. 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

63.  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

64.  
 ,(3
4
1
,)1(
)1(
)2(
2
1
,)2()43)(3(
2
)43()53(
2
)(
2
3
11
,
3
4
2
5
2
3
)(
2
3
1:where
,
12
)(and,
3
)(
4
2
2
3
2
2
1
42
987654
432
1
fgRgf
A
F
RgfR
A
F
ffgRfgRBA
A
RBfg
R
fgAF
fgRfgAF
FF
FFFFFF
FF
F
F
F
Tiijj































a
a
aa
64
Appendix: The Kuster-Toksöz values

65.   
  
 
 
 
ratio.aspectporeand)23(
1
)1(cos
)1(
,
43
3
,
3
,1),43()1(
),43)(1(35
2
)1(
2
21
),43(3539
4
2
),43)(1(11
),43(
3
4
2
2
2/121
2/32
9
8
7
6
5






























a
a
a
aaa
a
a


fg
f
K
R
K
K
BARBfRfRgAF
RfBR
f
R
g
RAF
RBfgfRgf
A
F
RfBfgRgAF
RBfgfgRAF
mm
m
m
f
65
Appendix: The Kuster-Toksöz values