1. Uncertainty in AVO: How can we measure it? Dan Hampson, Brian Russell
Hampson-Russell Software, Calgary
Maurizio Cardamone
ENI E&P Division, Milan, Italy

2. Overview
AVO Analysis is now routinely used for exploration and development.
But: all AVO attributes contain a great deal of “uncertainty” – there is a wide range of lithologies which could account for any AVO response.
In this talk we present a procedure for analyzing and quantifying AVO uncertainty.
As a result, we will calculate probability maps for hydrocarbon detection.

3. AVO Uncertainty Analysis: The basic process
STOCHASTIC AVO
MODEL G I
GRADIENT
INTERCEPT
BURIAL DEPTH
CALIBRATED:
FLUID
PROBABILITY
MAPS
PBRI
POIL
PGAS
AVO ATTRIBUTE
MAPS
ISOCHRON

4. “Conventional” AVO Modelling : Creating 2 pre-stack syntheticsGO IB GB
IN SITU = OIL
FRM = BRINE
Let us go one step back into the “conventional” flow of a typical AVO study.
It is quite a complex piece of technology, which is not yet well classified as it is data processing, modelling or interpretation.
Most probably it is a mixture of the three ingredients, with many different recipes available on the market.
On one side we have anyway the modelling - whenever a well is available - to provide the AVO analyst with a synthetic pre-stack CDP gather for assessment of the AVO behaviour and comparison with real CDP’s.
This starts with a set of well logs, preferably including a good quality S sonic measurement, even though this kind of a log can be quite effectively modelled from other logs.
It is common practice to start from this set of the so-called insitu and model different fluids, different saturation ratios, different porosity. This is normally carried out using the Biot-Gassmann equation.
Then you measure the modelled AVO responses and you can assess the sensitivity of AVO to the different petrphysical conditions, mainly to the different fluids (gas, oil, brine).

5. Monte Carlo Simulation: Creating many synthetics
I-G DENSITY FUNCTIONS
BRINE
OIL
GAS
GRADIENT
OCCURRENCE

6. The basic model Shale Sand Shale
We assume a 3-layer model with shale enclosing a sand (with various fluids).
Sand
Shale

7. The basic model Vp1, Vs1, r1 Vp2, Vs2, r2
The Shales are characterized by: P-wave velocity
S-wave velocity
Density
Vp1, Vs1, r1
Vp2, Vs2, r2

8. The basic model Vp1, Vs1, r1 Vp2, Vs2, r2
Each parameter has a probability distribution:
Vp1, Vs1, r1
Vp2, Vs2, r2

9. The basic model Shale Sand Shale The Sand is characterized by:
Brine Modulus
Brine Density
Gas Modulus
Gas Density
Oil Modulus
Oil Density
Matrix Modulus
Matrix density
Porosity
Shale Volume
Water Saturation
Thickness
Shale
Sand
Shale
Each of these has a probability distribution.

10. Trend Analysis Sand (Brine) Velocity
Some of the statistical distributions are determined from well log trend analyses:
Sand (Brine) Velocity

11. Determining distributions at selected locations.
Assume a Normal distribution. Get the Mean and Standard Deviation from the trend curves for each depth:

12. Trend Analysis: Other Distributions
Shale Velocity
Sand Density
Shale Density
Sand Porosity

13. Practically, this is how we set up the distributions:
Shale:
Vp Trend Analysis
Vs Castagna’s Relationship with % error
Density Trend Analysis
Sand:
Brine Modulus
Brine Density
Gas Modulus
Gas Density
Oil Modulus Constants for the area
Oil Density
Matrix Modulus
Matrix density
Dry Rock Modulus Calculated from sand trend analysis
Porosity Trend Analysis
Shale Volume Uniform Distribution from petrophysics
Water Saturation Uniform Distribution from petrophysics
Thickness Uniform Distribution

14. Calculating a single model response
Note that a wavelet is assumed known.
From a particular model instance, calculate two synthetic traces at different angles. 0o
45o
Top Shale
Sand
Base Shale

15. Calculating a single model response
Note that these amplitudes include interference from the second interface.
On the synthetic traces, pick the event corresponding to the top of the sand layer: 0o
45o
Top Shale P2
Sand P1
Base Shale

16. Calculating a single model response
Using these picks, calculate the Intercept and Gradient for this model:
I = P1
G = (P2-P1)/sin2(45) 0o
45o
Top Shale P2
Sand P1
Base Shale

17. Using Biot-Gassman substitution
Starting from the Brine Sand case, the corresponding Oil and Gas Sand models are generated using Biot-Gassmann substitution. This creates 3 points on the I-G cross plot:
BRINE
GAS
KGAS
GAS
OIL
KOIL
OIL G I

18. Brine Oil Gas Monte-Carlo Analysis
By repeating this process many times, we get a probability distribution for each of the 3 sand fluids: G
Brine
Oil
Gas I

19. The results are depth-dependent
Because the trends are depth-dependent, so are the predicted distributions:
@ 1000m
@ 1200m
@ 1400m
@ 1600m
@ 1800m
@ 2000m

20. The Depth-dependence can often be understood using Rutherford-Williams classification
Sand
Burial Depth
Impedance
Shale 1 2 3 4 5 6
Class 1
Class 2
Class 3

21. Bayes’ Theorem
Bayes’ Theorem is used to calculate the probability that any new (I,G) point belongs to each of the classes (brine, oil, gas):
where:
P(Fk) represent a priori probabilities and Fk is either brine, oil, gas;
p(I,G|Fk) are suitable distribution densities (eg. Gaussian) estimated from the stochastic simulation output.

22. How Bayes’ Theorem works in a simple case:
Assume we have these distributions:
VARIABLE
OCCURRENCE
Gas
Oil
Brine

23. How Bayes’ Theorem works in a simple case:
This is the calculated probability for
(gas, oil, brine).
VARIABLE
OCCURRENCE
100%
50%

24. When the distributions overlap, the probabilities decrease:
Even if we are right on the “Gas” peak, we can only be 60% sure we have gas.
VARIABLE
OCCURRENCE
100%
50%

25. Showing the effect of Bayes’ theorem
This is an example simulation result, assuming that the wet shale Vs and Vp are related by Castagna’s equation.

26. Showing the effect of Bayes’ theorem
This is an example simulation result, assuming that the wet shale Vs and Vp are related by Castagna’s equation.
This is the result of assuming 10% noise in the Vs calculation

27. Showing the effect of Bayes’ theorem
Note the effect on the calculated gas probability
1.0
0.5
0.0
Gas Probability
By this process, we can investigate the sensitivity of the probability distributions to individual parameters.

28. Example probability calculations
Oil
Brine
Gas

29. Real Data Calibration
In order to apply Bayes’ Theorem to (I,G) points from a real seismic data set, we need to “calibrate” the real data points.
This means that we need to determine a scaling from the real data amplitudes to the model amplitudes.
We define two scalers, Sglobal and Sgradient, this way:
Iscaled = Sglobal *Ireal Gscaled = Sglobal * Sgradient * Greal
One way to determine these scalers is by manually fitting multiple known regions to the model data.

30. Fitting 6 known zones to the model1 4 2 3 5 6

31. Real data example – West Africa
This example shows a real project from West Africa, performed by one of the authors (Cardamone).
There are 7 productive oil wells which produce from a shallow formation.
The seismic data consists of 2 common angle stacks.
The object is to perform Monte Carlo analysis using trends from the productive wells, calibrate to the known data points, and evaluate potential drilling locations on a second deeper formation.

32. One Line from the 3-D volume
Near Angle Stack
0-20 degrees
Far Angle Stack
20-40 degrees

33. One Line from the 3-D volume
Near Angle Stack
0-20 degrees
Shallow producing zone
Deeper target zone
Far Angle Stack
20-40 degrees

34. AVO Anomaly Near Angle Stack 0-20 degrees Far Angle Stack

35. Amplitude slices extracted from shallow producing zone
Near Angle Stack
0-20 degrees
+189
-3500
Far Angle Stack
20-40 degrees

36. Trend analysis Sand and Shale trends
Sand velocity
DENSITY
Sand density
VELOCITY
BURIAL DEPTH (m)
VELOCITY
Shale velocity
Shale density
BURIAL DEPTH (m)
DENSITY

37. Monte Carlo simulations at 6 burial depths
-1400
-1600
-1800
-2000
-2200
-2400

38. Near Angle amplitude map showing defined zones
Wet Zone 1
Well 6
Well 3
Well 5
Well 7
Well 1
Well 2
Wet Zone 2
Well 4

39. Calibration Results at defined locations
Wet Zone 1
Well 2
Wet Zone 2
Well 5

40. Calibration Results at defined locations
Well 6
Well 3
Well 4
Well 1

41. Using Bayes’ theorem at producing zone: oil
Near Angle Amplitudes
1.0
.80
.60
Probability of Oil
.30

42. Using Bayes’ theorem at producing zone: gas
Near Angle Amplitudes
1.0
.80
.60
Probability of Gas
.30

43. Near angle amplitudes of second event
Using Bayes’ theorem at target horizon
Near angle amplitudes of second event
1.0
.80
.60
Probability of oil on second event
.30

44. Verifying selected locations at target horizon

45. Summary
By representing lithologic parameters as probability distributions we can calculate the range of expected AVO responses.
This allows us to investigate the uncertainty in AVO predictions.
Using Bayes’ theorem we can produce probability maps for different potential pore fluids.
But: The results depend critically on calibration between the real and model data.
And: The calculated probabilities depend on the reliability of all the underlying probability distributions.