7 AVO Cross Plot

S ei e i s m i c I n v e r s i o n a n d AV AV O a p p l ie i e d t o L i t h o l o g i c P r ed ed i c t i o n   Part 7  – AV A V O Cr C r o s s p l o t t in in g

Introduction • In previous sections, we have looked at basic rock physics, post-stack inversion, P and S-wave recording and AVO modeling and analysis. • We used the Aki-Richards equations to perform both forward modeling and data analysis, extracting the intercept and gradient. • In this section, we will see how the crossplot of intercept against gradient can aid us in the interpretation of AVO anomalies anomalies.. • We will also link rock physics, AVO modeling and crossplotting, and show how this leads to the polarization analysis of AVO anomalies. 7-2

Introduction • In previous sections, we have looked at basic rock physics, post-stack inversion, P and S-wave recording and AVO modeling and analysis. • We used the Aki-Richards equations to perform both forward modeling and data analysis, extracting the intercept and gradient. • In this section, we will see how the crossplot of intercept against gradient can aid us in the interpretation of AVO anomalies anomalies.. • We will also link rock physics, AVO modeling and crossplotting, and show how this leads to the polarization analysis of AVO anomalies. 7-2

 AVO  A VO Crossplotting Crossplotting  AVO crossplotting  AVO crossplotting involves involves plotting the intercept intercept against against the gradient gradient and identifying identifying anomalies anomalies.. The theory of crossplotting was developed developed by Castagna el al (TLE, 1997, Geophysics, 1998) and Verm and Hilterman (TLE, 1995) and is based on two ideas: (1) The Rutherford/Wil Rutherford/Williams liams classification scheme. (2) The Mudrock line.  Although we discussed discussed the Rutherford/Willi Rutherford/Williams ams classification scheme in the last section, we will first briefly review the scheme. 7-3

Rutherford/Williams Classification Rutherford and Williams (1989) derived the following classification scheme for AVO anomalies, with further modifications by Ross and Kinman (1995) and Castagna (1997). The acoustic impedance changes refer to the anomalous layer: Class 1: Large increase in acoustic impedance. Class 2: Near-zero impedance contrast. Class 2p: Same as 2, with polarity change. Class 3: Large decrease in acoustic impedance. Class 4: Very large decrease in acoustic impedance coupled with small Poisson’s ratio change. 7-4

The Mudrock Line The mudrock line is a linear relationship between V P  and V S derived by Castagna et al (1985). The equation and original plot are shown below: V P  = 1.16 V S + 1360 m/sec = aV S + b

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The Mudrock Line Notice that this is n o t the same as a constant Poisson’s ratio, since this would be written as follows (without an intercept term):

V P  

2    2  2    1

V S  aV S

This will be illustrated in the next few slides, where a gas sand is shown below the mudrock line, and then lines of constant  are superimposed.

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The Mudrock Line 6000 5000 Mudrock Line

4000 3000 Gas Sand

VP (m/s) 2000 1000 0 0

1000

VS(m/s)

2000

3000

4000 7-7

The Mudrock Line 6000 5000  = 1/3 or Vp/Vs = 2

Mudrock Line

4000 3000 Gas Sand

VP (m/s) 2000 1000 0 0

1000

VS(m/s)

2000

3000

4000 7-8

The Mudrock Line 6000 5000  = 1/3 or Vp/Vs = 2

Mudrock Line

4000 3000 Gas Sand

VP (m/s) 2000 = 0.1 or Vp/Vs = 1.5

1000 0 0

1000

VS(m/s)

2000

3000

4000 7-9

Intercept versus Gradient • By using the Aki-Richards equation, Gardner’s equation, and the ARCO mudrock line, we can derive a simple relationship between intercept and gradient. Note that:

 A 

1   V P 

  

   2   V  p    

2 

2 

V S   V S V S      4   2   B , 2  V  p V P   V S V P      1  V P 

Gardner  :     aV 

0 .25  P 



     



1  V P  4 V P 

• If we assume that V P  / V S = c , a constant, we can show that:

9  B   A 1  2   5   c   4

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Intercept versus Gradient • Now let us use a few values of c and see how the previous equation simplifies. If c = 2, the most commonly accepted value, the gradient is the negative of the intercept (a -45 degree line on a crossplot):

 9 B   A1     A 5   4  4

• If c = 3, the gradient is zero, a horizontal line on the crossplot of intercept against gradient:

 9 B   A 1    0  5   9  4

• Various values of c produce the straight lines (“wet” trends) shown on intercept/gradient crossplots on the next page. 7-11

Mudrock lines on a crossplot for various V P/VS ratios (Castagna and Swan, 1998).

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Intercept / Gradient Crossplots • By letting c=2 for the background wet trend, we can now plot the various anomalous Rutherford and Williams classes (as extended by Ross and Kinman and Castagna et al). • Note that each of the classes will plot in a different part of the intercept/gradient crossplot area. • The anomalies form a roughly elliptical trend on the outside of the wet trend. • This is shown in the next figure.

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Gradient Base II P

Base

II

Base I Base

III

Top IV

Intercept Base IV Top III

Crossplot showing anomalies

Top I

“Wet” Trend Top II

Top II P

 V  p     2   V s   7-14

Example of crossplotting The following figures are taken from the first published example of AVO crossplotting:

(a) Cross-plot of well log derived A and B.

(b) Cross-plot of seismically derived A and B. Foster et al (1993)

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Intercept / Gradient Crossplot Here is an example of the crossplot in color:

(a) Uninterpreted crossplot.

(b) Interpreted crossplot, where the pink = top of gas, yellow = base of gas, and blue = hard

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Seismic Display from Int/Grad Xplots Note the validation of the previous results:

(a) Before interpretation

(b) After interpretation

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Cross-plot modeling • We will next consider a straightforward methodology for incorporating AVO crossplotting into AVO modeling. • This will provide us with a link between our discussion of fluid substitution with the Biot-Gassmann equations, and the crossplotting of AVO attributes from real data. • We will also discuss the effect of the wavelet on the crossplot, creating what other authors have termed the  AVO hodogram. • This article was written by Dr. Christopher Ross and appeared in the May-June 2000 issue of Geophysics. 7-18

The Proposed Modeling Flow • The modeling flow that will be used in this tutorial involves the following five steps: (1) Edit and prepare the well logs for AVO modeling. (2) Create fluid/lithology replacement logs. (3) Generate in-situ and fluid replacement AVO models. (4) Generate the appropriate AVO attributes for both models (e.g. Intercept and Gradient) (5) Crossplot the attributes from each model simultaneously.

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Well Logs Wireline well log suite for the  AVO modeling example, where the reservoir is annotated in yellow. The original shear wave log was created used multiple regression on the gammaray, SP and neutron porosity logs. Fluid replacement was done assuming a 40% water saturation in place of the original 100%. 7-20

1000 ft

Models Far offset = 20000 ft

The forward models from the wet and gas sand fluid substitution cases, using a full elastic wave-equation algorithm. The wiggle traces overlay the color amplitude envelope.

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 AVO responses from Model Example  AVO computations from the gas sand model of the previous slide, where the slide on the left shows an intercept x gradient product (A*B) and the slide on the right shows a weighted sum of the intercept and gradient. In the case, the weights are a = 0.5 and b = 0.31. (a) A*B plot

(b) a A+bB plot

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Fluid Vector Movement Fluid vector movement from the shale (top) to the wet sand (middle) to the gas-charged sand (bottom left). The colors now represent depth. These point come from the trough that occurs in the shaleover-sand interface, seen on the previous slide. 7-23

Crossplot of Model Example

(a) Simultaneous crossplot of the two models, in-situ=green points, and gas= purple points. The gray ellipse is the wet trend and the yellow/blue the gas.

(b) Trace display of the models, with crossplot colors superimposed. In-situ case on left and gas case on right. 7-24

Thickness and Bandwidth Effects • The crossplots in the next two slides represent the effects of thickness variations in the cleaner sand members of the modeled reservoir. • The first slide shows the unaltered case, a 50% reduction, and a 75% reduction, respectively. • Note the loss of definition as the sands are reduced in thickness. • The second slide shows the effect of seismic bandwidth change on the intercept and gradient. As the frequency is lowered, there is loss of definition.

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Effect of Sand Thickness

(a) Full crossplot through unaltered sand.

(b) Zoom of crossplot of the trough in the unaltered sand. 7-26

Effect of Sand Thickness

(a) Full crossplot through sand that has been thinned by 50%.

(b) Zoom of crossplot over trough in sand that has been thinned by 50%. 7-27

Effect of Sand Thickness

(a) Full crossplot through sand that has been thinned by 75%.

(b) Zoom of crossplot over trough in sand that has been thinned by 75%. 7-28

Effect of Sand Thickness

(a) Zoom of crossplot of the trough in the unaltered sand.

(b) Zoom of crossplot over trough in sand that has been thinned by 50%.

(c) Zoom of crossplot over trough in sand that has been thinned by 75%.

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Seismic Bandwidth Change

(a) Unfiltered (4/8-24/48 Hz) crossplot over over unaltered sand.

(b) Filtered 4/8-20/24 Hz) crossplot over over unaltered 7-30 sand.

 AVO polarization attributes • In the next part of this section, we will discuss the intercept/gradient hodogram and polarization attributes, first developed by Keho (“The AVO hodogram: Using polarization to identify anomalies”, presented at the 2000 SEG meeting in Calgary and published in TLE, November, 2001). • We will illustrate the concepts using both real and synthetic datasets.

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Gas Sand Example

Time Window

Top Gas Sand Base Gas Sand

We will first illustrate the hodogram using the gas sand anomaly above. This is an (A+B)/2, or pseudo-Poisson’s ratio plot. 7-32

The A-B crossplot

Here is the A-B crossplot of the points from trace 330 over the time window shown on the previous plot. There is no obvious anomaly

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The hodogram

B

A

time

Here is the hodogram, showing time as a third axis. Notice the extra information in the hodogram, and the clear anomaly at 630 ms.

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Polarization analysis • Rather than display the A and B attributes as a hodogram, we can compute the polarization angle from a running time window centered at time t on the attributes, as shown here: A

time = t 

B

Window length = M points

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Polarization analysis When we do this polarization analysis on trace 330, using a window length of 3 samples, the result is as seen to the left. Note the clear indication of an anomaly between 628 and 638 ms, at the known gas sand zone.

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Polarization angle attributes • The polarization angle, , is defined as positive upwards from the horizontal (A) axis. The result is highly dependent on the length of the running window. • The polarization angle difference by subtracting a background angle

is computed trend.

•  A third attribute is the polarization magnitude, which is defined as the RMS length of the cloud of points on the A-B crossplot. •  A fourth is the correlation coefficient squared. • The last is the polarization product, or the product of the magnitude and the polarization angle difference. 7-37

 A model example • Next, we will use a gas sand model, which was used by Christopher Ross in his paper “Comparison of popular AVO attributes, AVO inversion, and calibrated AVO predictions” (TLE, March, 2002), to illustrate the ideas just discussed. • The model gas sand is shown in the next slide. • The application of polarization analysis is shown in subsequent slides.

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M o d e l g as   s an d

Gas Sand

(a)

(b)

The plot above show the (a) crossplot zone analysis, and (b) sum of intercept and gradient (pseudo-Poisson’s ratio) for the gas sand anomaly.

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Effect of changing the window length • The next six slides show the effect of changing the running window length from 10 ms to almost 200 ms on the polarization angle difference. • In all cases, the removed trend was equal to 0, meaning that these plots are also equal to the polarization angle itself. • Notice that if the length of the window gets too large, the anomaly appears to move due to edge effects. 7-40

Model gas sand

Polarization difference with window = 10 ms.

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Model gas sand

Polarization difference with window = 18 ms.

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Model gas sand

Polarization difference with window = 30 ms.

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Model gas sand

Polarization difference with window = 62 ms.

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Model gas sand

Polarization difference with window = 102 ms.

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Model gas sand

Polarization difference with window = 182 ms.

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Polarization product • The next two slides show the polarization product for the gas sand example. • Recall that this is the product of the polarization angle difference and the polarization magnitude. •  Again, the trend angle is equal to 0. • Note that the result is slightly clearer than for the angle plots, but the anomaly was visible on either display. This is because the data is noise free.

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Model gas sand

Polarization product with window = 10 ms.

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Model gas sand

Polarization product with window = 30 ms.

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Colony sand example • Next, we will use a real data example. • This is a 2D line over a shallow gas sand in  Alberta (the Colony Colony sand). • The anomaly is a class 3 gas sand. • The sonic log from the discovery well is overlain at CDP 330. • The gas sand is at a time of 620 ms. • The next few slides show the result of polarization analysis. 7-50

Colony sand example

Polarization angle with window = 10 ms.

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Colony sand example

Polarization product with window = 10 ms.

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Colony sand example

Polarization angle with window = 18 ms.

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Colony sand example

Polarization product with window = 18 ms.

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Mahakam Delta example • In an article by Keho et al. in The Leading Edge, November, 2001, an interesting case study of polarization attributes is presented. • This study was done in the Mahakam Delta area of east Kalimantan. • The next slide shows aerial photo maps of the Mahakam Delta. • We will then look at the results of the polarization study. 7-55

Mahakam Delta example Herman Darman (Shell), F. Hasan Sidi (VICO), Agung Wiweko (Total), Bernard Lambert (Total), Bambang Seto (Total)

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Mahakam Delta example

 An amplitude slice showing a gas-filled incised valley, where the change to positive polarity (red) indicates the valley (Keho et al., 2001).

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Mahakam Delta example

 A vertical section showing the gas-filled incised valley. (Keho et al., 2001).

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Mahakam Delta example

 At top left is the intercept (A`) after rotation along the main trend of the crossplot, and at bottom right is the gradient (B`) perpendicular to the trend. See the figure in the upper right for the rotation. (Keho et al., 2001).

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Mahakam Delta example

 At top left is the “product indicator” (AB*), and at bottom right is the “delta product indicator” (  AB*). Both show detail in the channel but are quite noisy. (Keho et al., 2001). 7-60

Mahakam Delta example

Color convention used for representing the classes using polarization angle. (Keho et al., 2001).

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Mahakam Delta example

Polarization angle plot for the gas-filled channel. Notice that inside the channel, there appears to be a Class II anomaly (Keho et al., 2001).

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