Seismic Inversion and AVO applied to Lithologic Prediction

Part 7 AVO Crossplotting

IntroductionIn 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.We will also link rock physics, AVO modeling and crossplotting, and show how this leads to the polarization analysis of AVO anomalies.

AVO CrossplottingAVO crossplotting involves plotting the intercept against the gradient and identifying anomalies. The theory of crossplotting was developed by Castagna el al (TLE, 1997, Geophysics, 1998) and Verm and Hilterman (TLE, 1995) and is based on two ideas:

(1) The Rutherford/Williams classification scheme. (2) The Mudrock line.

Although we discussed the Rutherford/Williams classification scheme in the last section, we will first briefly review the scheme.

Rutherford/Williams ClassificationRutherford 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 Poissons ratio change.

The Mudrock LineThe mudrock line is a linear relationship between VP and VS derived by Castagna et al (1985). The equation and original plot are shown below:VP = 1.16 VS + 1360 m/sec = aVS + b

The Mudrock LineThis will be illustrated in the next few slides, where a gas sand is shown below the mudrock line, and then lines of constant s are superimposed.Notice that this is not the same as a constant Poissons ratio, since this would be written as follows (without an intercept term):

The Mudrock Line020002000400060001000300040000100030005000VP (m/s)VS(m/s)Mudrock LineGas Sand

The Mudrock Line020002000400060001000300040000100030005000VP (m/s)VS(m/s)Mudrock LineGas Sand = 1/3 orVp/Vs = 2

The Mudrock Line020002000400060001000300040000100030005000VP (m/s)VS(m/s)Mudrock LineGas Sand = 1/3 orVp/Vs = 2 = 0.1 orVp/Vs = 1.5

Intercept versus GradientBy using the Aki-Richards equation, Gardners equation, and the ARCO mudrock line, we can derive a simple relationship between intercept and gradient. Note that: If we assume that VP / VS = c, a constant, we can show that:

Intercept versus GradientNow 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):If c = 3, the gradient is zero, a horizontal line on the crossplot of intercept against gradient:Various values of c produce the straight lines (wet trends) shown on intercept/gradient crossplots on the next page.

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

Intercept / Gradient CrossplotsBy 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.

GradientInterceptWet TrendBase IIIBase IIBase II PTop IVTop IIITop IITop II PBase IVTop IBase ICrossplot showing anomalies

Example of crossplottingFoster et al (1993)(a) Cross-plot of well logderived A and B.(b) Cross-plot of seismicallyderived A and B.The following figures are taken from the first published example of AVO crossplotting:

Intercept / Gradient Crossplot(b) Interpreted crossplot, where the pink = top of gas, yellow = base of gas, and blue = hard streak.(a) Uninterpreted crossplot.Here is an example of the crossplot in color:

Seismic Display from Int/Grad Xplots(a) Before interpretation(b) After interpretationNote the validation of the previous results:

Cross-plot modelingWe 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.

The Proposed Modeling FlowThe 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.

Well LogsWireline well log suite for the AVO modeling example, where the reservoir is annotated in yellow. Theoriginal shear wave log wascreated used multipleregression on the gamma-ray, SP and neutron porosity logs. Fluid replacement was done assuming a 40% water saturation in place of the original 100%.

ModelsWet SandGas Sand1000 ftFar offset = 20000 ft The forward modelsfrom the wet and gassand fluid substitutioncases, using a fullelastic wave-equationalgorithm. The wiggle traces overlaythe color amplitudeenvelope.

AVO responses from Model ExampleAVO computationsfrom the gas sand model of the previous slide, where the slideon the left shows anintercept x gradient product (A*B) and theslide on the right showsa weighted sum of theintercept and gradient.In the case, the weightsare a = 0.5 and b = 0.31.(a) A*B plot(b) aA+bB plot

Fluid Vector MovementFluid vector movementfrom the shale (top) tothe wet sand (middle)to the gas-charged sand (bottom left). Thecolors now representdepth. These pointcome from the troughthat occurs in the shale-over-sand interface,seen on the previousslide.

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.

Thickness and Bandwidth EffectsThe 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.

Effect of Sand Thickness(a) Full crossplot throughunaltered sand.(b) Zoom of crossplot of thetrough in the unaltered sand.

Effect of Sand Thickness(a) Full crossplot throughsand that has been thinnedby 50%.(b) Zoom of crossplot over trough in sand that has been thinned by 50%.

Effect of Sand Thickness(a) Full crossplot throughsand that has been thinnedby 75%.(b) Zoom of crossplot over trough in sand that has been thinned by 75%.

Effect of Sand Thickness(c) Zoom of crossplot over trough in sand that has been thinned by 75%.(b) Zoom of crossplot over trough in sand that has been thinned by 50%.(a) Zoom of crossplot of the trough in the unaltered sand.

Seismic Bandwidth Change(a) Unfiltered (4/8-24/48 Hz)crossplot over over unalteredsand.(b) Filtered 4/8-20/24 Hz)crossplot over over unalteredsand.

AVO polarization attributesIn 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.

Gas Sand ExampleWe will first illustrate the hodogram using the gas sand anomaly above. This is an (A+B)/2, or pseudo-Poissons ratio plot.

The A-B crossplotHere 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.

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

Polarization analysisRather 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:

Polarization analysisWhen 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.

Polarization angle attributesThe polarization angle, f, 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 Df is computed by subtracting a background angle ftrend.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.

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.

Model gas sandThe plot above show the (a) crossplot zone analysis, and (b) sum of intercept and gradient (pseudo-Poissons ratio) for the gas sand anomaly.(a)(b)Gas Sand

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.

Polarization difference with window = 10 ms.Model gas sand

Model gas sandPolarization difference with window = 18 ms.

Model gas sandPolarization difference with window = 30 ms.

Model gas sandPolarization difference with window = 62 ms.

Model gas sandPolarization difference with window = 102 ms.

Model gas sandPolarization difference with window = 182 ms.

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.

Polarization product with window = 10 ms.Model gas sand

Model gas sandPolarization product with window = 30 ms.

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 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.

Polarization angle with window = 10 ms.Colony sand example

Polarization product with window = 10 ms.Colony sand example

Colony sand example Polarization angle with window = 18 ms.

Colony sand example Polarization product with window = 18 ms.

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.

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

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).

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

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).

Mahakam Delta example At top left is the product indicator (AB*), and at bottom right is the delta product indicator (DAB*). Both show detail in the channel but are quite noisy. (Keho et al., 2001).

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

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).

ConclusionsIn this section, we first discussed the concept of AVO crossplotting.We then showed how fluid replacement modeling and cross-plot analysis could be combined.For this analysis, we used a GOM dataset from a paper by Chris Ross.We then looked at a new concept in cross-plot analysis, called polarization analysis, or the AVO hodogram.We illustrated polarization analysis using a model dataset and real datasets from Alberta and Indonesia.