CGG GeoSoftware
AFI Theory
AFI Theory
Contents
AFI Model Description
Designing Probability Distributions for Parameters
AFI Simulation Theory
Biot-Gassmann Calculations for AFI
Probability Calculations in the AFI Program
Bayes Theorem
Scaling
AFI Model Description
This model is used in the AFI program. The basic model used in AFI is a 3-layer model,
consisting of a sand enclosed by shales. The shales are assumed to be wet (of course), while the
sand is modeled with brine, oil, and gas alternately. Since we will be using this model to analyze
AVO responses, the parameters describing each of the layers must define the P-wave velocity, Swave velocity, and density for the layer. In fact, AFI describes the shales directly with these
parameters:
May 2019
1
CGG GeoSoftware
AFI Theory
Each of these parameters is defined as a probability distribution, which determines the relative
spread of values expected for that parameter. The figure above shows a Normal or Gaussian
distribution for the shale density. Note that the shales above and below the sand are assumed to
have the same distributions.The available distributions are:
The Sand layer also has rock physics parameters, as shown below:
Each of these parameters can be described by a probability distribution, although in practice,
many are set as constants. These rock physics parameters are used in the Biot-Gassmann
substitution to calculate the effects of changing fluids within the sand layer.
A stochastic model is a particular set of parameters for each of the layers in the model. Note
that in practice, we may have different stochastic models for different depths. This is because
the compaction trends for a particular area cause the rock properties to vary with depth. AFI can
define and store stochastic models at any number of depth levels.
1. The first step in an AFI project is to define the stochastic models for a series of depths (so
these depths become control levels). This is usually done by Trend Analysis of well data within
the area, assuming a normal distribution and deriving the mean and standard deviation. Note that
the samples must be from corresponding lithology, preferably the same formation.
In practice, we derive values as below:
Shale
May 2019
VP:
Trend Analysis
VS:
Castagna's Relationship with % error
density (ρ):
Trend Analysis
2
CGG GeoSoftware
Sand
AFI Theory
Brine Modulus:
Constants for the area
Brine ρ:
Constants for the area
Gas Modulus:
Constants for the area
Gas ρ:
Constants for the area
Oil Modulus:
Constants for the area
Oil ρ:
Constants for the area
Matrix Modulus:
Constants for the area
Matrix ρ:
Constants for the area
Dry Rock Modulus:
Calculated from sand trend analysis
Porosity (f):
Trend Analysis
Shale Volume:
Uniform distribution from petrophysics
Water Saturation:
Uniform distribution from petrophysics
Thickness:
Uniform distribution
2. For the basic shale-sand-shale model, calculate two synthetic traces at different offset angles.
We assume that the wavelet is known. One trace should be at 0° offset.
3. Pick the event representing the sand layer. Then determine the Intercept and Gradient:
Pick at 0° is the Intercept in msec, and the Gradient is:
(Offset pick - 0° pick)/sin2(angle of 2nd offset).
4. Then substitute Oil and Gas using the Biot-Gassmann process and plot all of these on the
Intercept-Gradient cross plot.
5. Repeat until we have a probability distribution. See AFI Simulation Theory.
Each dot represents the gradient and intercept values derived from a model created through the
probability distributions of the parameters.
May 2019
3
CGG GeoSoftware
AFI Theory
Designing Probability Distributions for Parameters
We recommend graphing the data known from the local wells for trend analyses. For example,
you can graph the value versus frequency for porosity, density and dry rock moduli. These
parameters are usually plotted against depth to show depth-dependent trends. Some of these
values will come from log analyses, some from velocity surveys and some from core studies.
There will also be information available for regional values. These would apply to the
parameters usually treated as constants.
Some values would be measured from the well logs and mapped out. Then the maps would
indicate the value. Sometimes a value usually determined by trend analysis has instead been
mapped (e.g., porosity).
You must ensure that the values you are using apply to the formations being studied and not to a
different formation that is either at a different depth or has different lithology. You must also
ensure that the distribution is not showing unrealistic or impossible values (though our program
will not use impossible values), as shown below.
AFI Simulation Theory
The process consists of the following steps. First, the program picks the rock physics parameter
values from the corresponding probability distributions to create a 3-layer model. Then, a
particular instance of that model is created for the brine case. This uses the Biot-Gassmann
theory described in the previous section, Biot-Gassmann Calculations. Using that model and that
fluid case (brine), two synthetic traces are generated internally, as shown in this figure:
May 2019
4
CGG GeoSoftware
AFI Theory
Note that for this process, we must already have a wavelet. From the calculated synthetic traces,
the top of the sand (or alternatively, the base) is picked automatically. From the picked
amplitudes, the theoretical intercept and gradient are calculated. Note that, although only the top
of the sand is picked, the sand thickness is still modeled because of wavelet interference from the
base.
This creates a single point in the Intercept/Gradient plot, corresponding to a brine case. Now,
AFI repeats the process two more times using Biot-Gassmann calculations to replace the brine
with oil and then with gas. At this point, there are three values in an Intercept/Gradient plot for
that one model. Finally, this 3-step process is repeated many times for each depth level, creating
a new rock parameter model each time. Typically 200 models would have been created, giving
600 points. This creates a cluster in an Intercept/Gradient plot for each of the fluid types at each
depth level.
These models are created for various depths, and as the parameters are usually depth-dependent,
so are the resulting distributions. Therefore, the AVO plots are also depth-dependent, as shown
below.
Biot-Gassmann Calculations for AFI
This section outlines the mathematical algorithms used to apply fluid replacement modeling to
the input wet sand layer to create the corresponding gas and oil filled cases. The theory for this
section is detailed in Mavko et al (The Rock Physics Handbook), and is based on the theory of
May 2019
5
CGG GeoSoftware
AFI Theory
Biot and Gassmann. See the AFI program for the practical application. See also the AVO BiotGassmann Theory pdf.
The basic problem starts with a base case sand layer. Usually, this layer is wet, but we can
accommodate the possibility of hydrocarbons as well. The objective is to calculate Vp, Vs, and
density for the case where the brine is replaced by oil and gas. During this conversion process,
most of the Biot-Gassmann parameters are derived from stochastic variables, which means they
take on random values. This input layer is defined by the following constant parameters:
VSHALE0 = volume of shale in base sand layer
ρ0 = density of base sand layer
SW0 = water saturation of base sand layer (usually = 1.0)
ρH0 = density of hydrocarbon in base sand layer
ρW0 = density of water in base sand layer
f0 = porosity of base sand layer
VP0 = P-wave velocity of base sand layer
KW0 = water bulk modulus in base sand layer
KH0 = hydrocarbon bulk modulus in base sand layer
KM0 = matrix bulk modulus in base sand layer
All of these parameters are listed on the Sand Parameters page of the Model Parameters
window, and are normally determined from trend analysis of well log data.
Step 1:Calculate and store certain base sand parameters
The first step is to calculate three new parameters, which are stored for all later calculations.
Calculate ρM0 = Matrix density of base sand layer.
ρf0 = fluid density = ρw0 x Sw0 + ρH0 x (1 – SW0)
𝜌𝜌𝑀𝑀0 =
𝜌𝜌𝑀𝑀0 − 𝜌𝜌𝑓𝑓0 × 𝜙𝜙0
1 − 𝜙𝜙0
Calculate σDRY = dry rock Poisson's ratio, as measured for a rock in a laboratory vacuum. This
value is calculated from the base VSHALE value using the empirical formula contained in this
plot:
May 2019
6
CGG GeoSoftware
AFI Theory
Calculate KB0 = dry rock Bulk Modulus:
Kf =
1
 SW0

 KW
 0
S = 3*
(
 1 − SW0
+

K H0

)
(1 − σ DRY )
(1 + σ DRY )
M = VP20 * ρ0
a= S − 1
 KM0 
−S + M
b = f0 * S * 
 K −1
KM0
 f



M   K M 0
*
− 1
c = −f0 *  S −


K M 0   K f


y=
(− b +
b2 − 4ac
2a
)
K B 0= (1 − y ) * K M 0
The values of ρM0, νDRY, and KB0 are calculated and printed on the Sand Parameters page when
the Calculate button is pressed:
May 2019
7
CGG GeoSoftware
AFI Theory
Step 2: Calculate the new fluid replaced values for VP, VS, and Density
The previous step is only performed once, based on the base sand parameters. The next step is
performed once for each realization generated from the stochastic model.
The input values for this step are:
ρM0 = Matrix density of base sand layer (calculated in Step 1)
νDRY = dry rock Poisson's ratio (calculated in Step 1)
KB0 = dry rock Bulk Modulus (calculated in Step 1)
ρW = output water density (from stochastic model)
Sw = output water saturation (from stochastic model)
f = output porosity (from stochastic model)
ρH = output hydrocarbon density (from stochastic model)
ρM = output matrix density (from stochastic model)
KW = output water bulk modulus (from stochastic model)
KH = output hydrocarbon bulk modulus (from stochastic model)
The algorithm is:
KP =
φ0
 1
1 


−
K
K
M0 
 B0
ρ= ρW SW φ + ρ H (1 − SW )φ + ρ M (1 − φ ) (the desired output density)
Kf =
KB =
1
 SW [1 − SW ] 


+
K H 
 KW
1
 φ
1 


+
 KP KM 
May 2019
8
CGG GeoSoftware
𝜇𝜇𝐵𝐵 =
VP
2
AFI Theory
3𝐾𝐾𝐵𝐵 3�1 − 𝜈𝜈𝑑𝑑𝑑𝑑𝑑𝑑 �
�
− 1�
4
1 + 𝜈𝜈𝑑𝑑𝑑𝑑𝑑𝑑


2



KB 


1 −


K
4 µB 
M 

KB +
+

3

 1 − φ − K B 
 
K M  φ 
+

KM
K φ 

=
ρ
2
VS =
µB
ρ
(the desired output P-wave velocity)
(the desired output S-wave velocity)
Probability Calculations in the AFI Program
This section outlines the mathematical theory used to perform probability calculations in AFI.
The basic problem is this: We have three clusters of points, corresponding to brine, oil, and gas.
Given a new point, calculate the probability that this point belongs to each of the three clusters.
We assume that each of the clusters can be represented by a 2-dimensional probability
distribution. Consider, for example, the Gas distribution. Assuming that Gas was the only
cluster present, we could write
Pgas ( I , G ) = probability that the point (I,G) is a gas.
There are many possible alternatives for the theoretical probability distribution. The one which
AFI uses is the Cauchy distribution.
Define the vector x as the 2-dimensional vector at the desired location:
 x1   I 
=
x =
  
 x2  G 
May 2019
(1)
9
CGG GeoSoftware
AFI Theory
We also have the set of points in the gas cluster:
 x1k   I k 
=
x k =
=
the location of the kth point in the cluster
k
k
 x2  G 
(2)
The vector µ is the vector containing the mean values of the cluster points along the I and G
directions for the gas cluster:
1
 µ1   I gas   N
µ =
=
 
  =
µ
G
 2   gas   1
N



k =1

N

x2k 
∑
k =1

N
∑x
k
1
(3)
The matrix Σ is the covariance matrix for the cluster:
σ 12 
σ
Σ =  11

σ 12 σ 22 
(4)
where:
σ ij=
1
N
N
∑ (x
k =1
k
i
− µi )( x kj − µ j )
(5)
Now define the Mahalanobis distance as:
∆ 2 = ( x − μ ) T Σ −1 ( x − μ )
(6)
The Mahalanobis measurement accounts for correlations and is not dependent on the scale. It
estimates the ellipsoid of the probability distribution through covariances, takes the distance of
the point in question from the center of mass for the distribution and divides that by the width of
the ellipsoid in the direction of the point.
We can write that the probability associated with the point x, given a single gas cluster is:
pgas ( x ) =
1
2p Σ
1/ 2
(1 + ∆ 2 )
(7)
Equation (7) is the 2-dimensional "Cauchy" distribution, and this is the probability distribution
which is plotted in color on the Simulation Display Maps.
Bayes Theorem
To complete the process, we need to account for the fact that there are, in fact, three
distributions. The theory which allows us to handle this is Bayes' theorem, written for this case
as:
P ( gas | I , G ) =
May 2019
Pgas ( I , G ) * Pgas
 Pgas ( I , G ) * Pgas + Poil ( I , G ) * Poil + Pbrine ( I , G ) * Pbrine 
10
(8)
CGG GeoSoftware
AFI Theory
In equation (8)
P ( gas | I , G ) = the probability of gas at point (I,G) given all three clusters
Pgas ( I , G ) = the probability of gas at point (I,G) given the gas cluster alone (calculated
previously)
Pgas = the prior probability of gas, i.e., the probability before doing any analysis. Within AFI,
the prior probabilities for each of the fluids are assumed to be 1/3, i.e., no prior information.
For this reason, the preceding equation reduces to:
P ( gas | I , G ) =
Pgas ( I , G )
 Pgas ( I , G ) + Poil ( I , G ) + Pbrine ( I , G ) 
(9)
There are similar equations for the probability of oil and brine. Note that the denominator is the
same for all the equations.
The ability to distinguish between the fluids depends on the probability distributions and their
overlaps. Broad distributions will overlap and create uncertainty, even for points that plot on the
peaks of the distributions, since the overlap means that there still is a chance of the sample
having a different fluid, as shown below.
The basic form of the Bayes theorem is:
P (H | C ) =
P(H | C )P(H )
or posterior probability=(conditional times prior)/normalizer
P(C )
The probability of "H" (i.e. an "hypothesis": do we have gas?) for a specific condition (such as at
a geographical point (x, y) or given a condition C) is the probability of H occurring without
knowing the condition (this is the prior probability) times the probability of H occurring given
the condition (the conditional probability), all divided by the prior probability of the condition
occuring. For our case, the condition (x,y) includes all possibilities for the point (x,y), hence it is
the probability of gas, oil or brine being present. This normalizes the equation, and is 100% in
many calculations.
May 2019
11
CGG GeoSoftware
AFI Theory
Scaling
To use Bayes' Theorem to a real data set, we must calibrate the real data points, i.e. determine the
scaling required that will match the model amplitudes, Iscaled (for the intercept) and Gscaled (for the
gradient) to the real data amplitudes Ireal and Greal. We use two scalers, Sglobal and Sgradient.
Iscaled = Sglobal * Ireal
Gscaled = Sglobal * Sgradient * Greal
May 2019
12