Uncertainty in AVO:

How can we measure it?

Dan Hampson, Brian RussellHampson-Russell Software,

Calgary

Maurizio CardamoneENI E&P Division, Milan, Italy

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.

AVO Uncertainty Analysis:The basic process

AVO AVO ATTRIBUTEATTRIBUTEMAPSMAPSISOCHRONISOCHRONMAPSMAPS

GRADIENTGRADIENT INTERCEPTINTERCEPT BURIAL BURIAL DEPTHDEPTH

CALIBRATED:CALIBRATED:

STOCHASTIC STOCHASTIC AVOAVOMODELMODEL

GG

IIFLUIDFLUID

PROBABILITYPROBABILITY MAPSMAPS

PPBRIBRI

PPOILOIL

PPGASGAS

“Conventional” AVO Modelling : Creating 2 pre-stack synthetics

IO GO

IB GB

IN SITU = OILIN SITU = OIL

FRM = BRINEFRM = BRINE

Monte Carlo Simulation: Creating many synthetics

0

25

50

75

I-G DENSITY FUNCTIONS I-G DENSITY FUNCTIONS

BRINE OIL GAS

We assume a 3-layer model We assume a 3-layer model with shale enclosing a sand with shale enclosing a sand (with various fluids).(with various fluids).

Shale

Shale

Sand

The basic model

The The ShalesShales are characterized are characterized by:by:

P-wave velocity P-wave velocity

S-wave velocityS-wave velocity

DensityDensity

Vp1, Vs1, 1

Vp2, Vs2, 2

The basic model

Each parameter has a Each parameter has a probability distribution:probability distribution:

Vp1, Vs1, 1

Vp2, Vs2, 2

The basic model

The The SandSand is is characterized by:characterized by:

Brine ModulusBrine Modulus

Brine DensityBrine Density

Gas ModulusGas Modulus

Gas DensityGas Density

Oil ModulusOil Modulus

Oil DensityOil Density

Matrix ModulusMatrix Modulus

Matrix densityMatrix density

PorosityPorosity

Shale VolumeShale Volume

Water Water SaturationSaturation

ThicknessThickness

Each of these has a Each of these has a probability distribution.probability distribution.

Shale

Shale

Sand

The basic model

Sand (Brine) Velocity

0

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

0.4 0.9 1.4 1.9 2.4 2.9 3.4DBSB (Km)

Some of the statistical distributions are determined from well log trend analyses:

Trend Analysis

Determining distributions at selected locations.

0

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

0.4 0.9 1.4 1.9 2.4 2.9 3.4DBSB (Km)

Assume a Normal distribution. Get the Mean and Standard Deviation from the trend curves for each depth:

Trend Analysis: Other Distributions

0

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

0.4 0.9 1.4 1.9 2.4 2.9 3.4DBSB (Km)

Shale Velocity

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

2.8

3.0

0.4 0.9 1.4 1.9 2.4 2.9 3.4DBSB (Km)

Sand Density

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

2.8

3.0

0.4 0.9 1.4 1.9 2.4 2.9 3.4DBSB (Km)

0%

5%

10%

15%

20%

25%

30%

35%

40%

0.4 0.9 1.4 1.9 2.4 2.9 3.4DBSB (Km)

Shale Density

Sand Porosity

Shale:Shale:

VVpp Trend AnalysisTrend Analysis

VVss Castagna’s Relationship with % Castagna’s Relationship with % errorerror

DensityDensity Trend AnalysisTrend Analysis

Sand:Sand:

Brine ModulusBrine Modulus

Brine DensityBrine Density

Gas ModulusGas Modulus

Gas DensityGas Density

Oil ModulusOil Modulus Constants for the area Constants for the area

Oil DensityOil Density

Matrix ModulusMatrix Modulus

Matrix densityMatrix density

Dry Rock Modulus Dry Rock Modulus Calculated from sand trend analysis Calculated from sand trend analysis

PorosityPorosity Trend Analysis Trend Analysis

Shale VolumeShale Volume Uniform Distribution from Uniform Distribution from petrophysicspetrophysics

Water SaturationWater Saturation Uniform Distribution from petrophysics Uniform Distribution from petrophysics

ThicknessThickness Uniform Distribution Uniform Distribution

Practically, this is how we set up the distributions:

Top Shale

Base Shale

Sand

From a particular model instance, calculate two synthetic traces at different angles.

0o 45o

Note that a wavelet is assumed known.

Calculating a single model response

Top Shale

Base Shale

Sand

0o 45o

On the synthetic traces, pick the event corresponding to the top of the sand layer:

P1

P2

Note that these amplitudes include interference from the second interface.

Calculating a single model response

Top Shale

Base Shale

Sand

0o 45o

P1

P2

Using these picks, calculate the Intercept and Gradient for this model:

I = P1

G = (P2-P1)/sin2(45)

Calculating a single model response

GI

GI

GI

OILOIL

KKOILOIL

OILOIL

GASGAS

KKGAGA

SS

GASGAS

BRINEBRINE

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:

Using Biot-Gassman substitution

I

G

BrineOilGas

By repeating this process many times, we get a probability distribution for each of the 3 sand fluids:

Monte-Carlo Analysis

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

-0.8 -0.6 -0.4 -0.2 0 0.2 0.4

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

-0.8 -0.6 -0.4 -0.2 0 0.2 0.4

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

-0.8 -0.6 -0.4 -0.2 0 0.2 0.4

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

-0.8 -0.6 -0.4 -0.2 0 0.2 0.4

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

-0.8 -0.6 -0.4 -0.2 0 0.2 0.4

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

-0.8 -0.6 -0.4 -0.2 0 0.2 0.4

@ 1000m@ 1000m @ 1200m@ 1200m @ 1400m@ 1400m

@ 1600m@ 1600m @ 1800m@ 1800m @ 2000m@ 2000m

Because the trends are depth-dependent, so are the predicted distributions:

The results are depth-dependent

The Depth-dependence can often be understood using Rutherford-Williams

classification

SandSand

Burial DepthBurial Depth

Imp

edan

ceIm

ped

ance ShaleShale

1

1

2

2

3

3

4

4

5

5

6

6

Class 3

Class 2 Class 1

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.

k kk FPFGIp

FPFGIpGIFP

*,

)~(*

~,

,~

How Bayes’ Theorem works in a simple case:

VARIABLEVARIABLE

OC

CU

RR

EN

CE

OC

CU

RR

EN

CE

Assume we have these distributions:

GasOil

Brine

VARIABLEVARIABLE

OC

CU

RR

EN

CE

OC

CU

RR

EN

CE

100%

50%

This is the calculated probability for (gas, oil, brine).

How Bayes’ Theorem works in a simple case:

When the distributions overlap, the probabilities decrease:

VARIABLEVARIABLE

OC

CU

RR

EN

CE

OC

CU

RR

EN

CE

100%

50%

Even if we are right on the “Gas” peak, we can only be 60% sure we have gas.

This is an example simulation result, assuming that the wet shale Vs and Vp are related by Castagna’s equation.

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

Showing the effect of Bayes’ theorem

Note the effect on the calculated gas probability

0.0

0.5

1.0

Gas Probabilit

y

By this process, we can investigate the sensitivity of the probability distributions to individual parameters.

Showing the effect of Bayes’ theorem

Example probability calculations

Gas

Oil Brine

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.

Fitting 6 known zones to the model

1

4

2

3

56

1

4

2

3

56

1 2

4 5 6

3

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.

Near Angle Stack0-20 degrees

Far Angle Stack20-40 degrees

One Line from the 3-D volume

Near Angle Stack0-20 degrees

Far Angle Stack20-40 degrees

Shallow producing zone

Deeper target zone

One Line from the 3-D volume

Near Angle Stack0-20 degrees

Far Angle Stack20-40 degrees

AVO Anomaly

Amplitude slices extracted from shallow producing zone

Near Angle Stack0-20 degrees

Far Angle Stack20-40 degrees

-3500

+189

Trend analysisSand and Shale trends

1000

1500

2000

2500

3000

3500

4000

4500

5000

500 700 900 1100 1300 1500 1700 1900

VELO

CIT

Y

1.50

1.75

2.00

2.25

2.50

2.75

3.00

500 700 900 1100 1300 1500 1700 1900

DEN

SIT

Y

1000

1500

2000

2500

3000

3500

4000

500 700 900 1100 1300 1500 1700 1900 2100 2300 2500

BURIAL DEPTH (m)

VELO

CIT

Y

1.50

1.75

2.00

2.25

2.50

2.75

3.00

500 700 900 1100 1300 1500 1700 1900

BURIAL DEPTH (m)

DEN

SIT

Y

Sand velocity

Shale velocity

Sand density

Shale density

Monte Carlo simulations at 6 burial depths

-1400 -1600 -1800

-2000 -2200 -2400

Near Angle amplitude map showing defined zones

Wet Zone 1

Wet Zone 2

Well 6

Well 7

Well 3 Well 5Well 1

Well 2

Well 4

Calibration Results at defined locations

Wet Zone 1

Wet Zone 2

Well 2

Well 5

Well 3

Well 4

Well 6

Well 1

Calibration Results at defined locations

.30

.60

1.0

Probability of Oil

.80

Near Angle Amplitudes

Using Bayes’ theorem at producing zone: oil

.30

.60

1.0

Probability of Gas

.80

Near Angle Amplitudes

Using Bayes’ theorem at producing zone: gas

Near angle amplitudes of second event

.30

.60

1.0

.80

Probability of oil on second event

Using Bayes’ theorem at target horizon

Verifying selected locations at target horizon

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.