Extended Elastic Impedance Using HRS-9

Extended Elastic Impedance using HRS-9 Brian Russell Hampson-Russell, A CGGVeritas Company

Introduction  In a recent SEG Distinguished Lecture, Patrick Connolly outlined BP’s company-wide approach to fluid and lithology prediction using seismic data.  The cornerstones of this approach are the Coloured Inversion (CI) and Extended Elastic Impedance (EEI) methods.  This talk will first present a general framework for pre-stack and post-stack inversion methods.  I will then review the principles of EEI inversion within this framework.  Finally, I will show how the EEI method has been implemented in HRS-9, using both model-based and coloured inversion 2

Inversion in general Geological Model

Seismic Volume

Inversion Algorithm Inverted Seismic Volume The above flowchart shows the general approach to seismic trace inversion, which involves a geological model, a seismic volume and an inversion algorithm. We can apply the method to either pre- or post-stack seismic data.

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Post-stack inversion AI Model Volume

Stacked Volume Inversion Algorithm AI Inversion

The earliest trace inversion approach involved building an AI (acoustic impedance, or rVP) model and inverting the stacked volume to create an AI output. 4

Model building  Building a model volume involves the following steps:  Create the log property of interest at each well location and insert it into the model (in this case, we create acoustic impedance by multiplying VP by density).  Make sure the well logs match the seismic data in time by performing correlation with an extracted wavelet.  Interpolate the logs using an algorithm such as inverse-distance weighting or kriging.  Insert the seismic picks to guide the interpolation structurally.  Apply a low pass filter (typically 0 – 15 Hz) so that the detail in the inversion will come from the seismic data. 5

Types of inversion algorithms These post-stack inversion methods are available in HRS-9: Recursive:

Bandlimited inversion, in which the seismic trace is integrated and added to the low frequency part of the model. Model Based: Iteratively updates the initial model to find a best fit to the synthetic. Sparse Spike: Constrained to produce as few events as possible, with the low frequency model added in. Coloured: Spectrum of seismic data is shaped to the well log spectrum and a 90 degree phase shift applied. In the standard implementation, no low frequencies are added back (relative impedance) but in our implementation we can add them back (absolute impedance). 6

Gas sand stack

For example, here is a stack over a gas sand from Alberta, showing a “bright-spot” anomaly. 7

Post-stack inversion Here is the acoustic impedance inversion of the previous stack. The gas sand is a low impedance event, which is ambiguous because the shales are also low impedance. AI

Note that the seismic reflectivity can be written as the AI difference divided by twice its average:

AI RAI = 2 AI

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Pre-stack simultaneous inversion AI, SI and r models

Angle gathers

Simultaneous inversion AI, SI and density volumes

A more recent inversion approach is to build AI (rVP), SI (rVS) and density (r) models and invert the seismic angle gathers simultaneously. The next two slides show the gathers and the pre-stack inversion result.

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The seismic gathers Here is a portion of a 2D seismic line showing the gas sand “bright-spot”.

The seismic line is the “stack” of a series of CMP gathers, as shown here. The gas sand is a typical Class 3 AVO anomaly.

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Pre-stack simultaneous inversion AI

Vp/Vs

Note the unambiguous low Vp/Vs ratio at the gas sand.

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Near and far trace stacks Near angle (0-15o) stack

Far angle (15-30o) stack

A very robust AVO method involves analyzing near and far angle stacks, as shown here. Note the amplitude of the “bright-spot” event is stronger on the far-angle stack than it is on the near-angle stack. But what does this mean?

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Elastic Impedance  The equivalent impedance method to near and far angle stacking is Elastic Impedance, or EI (Connolly,1999).  To understand EI, recall the Aki-Richards equation:

 VS  VP r b R P ( )  a c , where : 2V P 2VS 2r a  1  tan 2  , b  8 K sin 2  , and c  1  4 K sin 2  .  Connolly postulated that associated with this equation is an underlying elastic impedance, written (where I have renamed the EI reflectivity to match the AI concept):

1  EI ( ) 1 R EI ( )    ln EI ( ), where EI ( )  V PaV Sb r c 2 EI ( ) 2 13

Elastic impedance inversion EI() Model Volume

Angle Stack Volume Inversion Algorithm EI() Inversion

The inversion approach for EI involves building an EI() model and inverting an angle stack volume at an angle  to create an EI output.

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Gas sand case study Here is the comparison between the EI inversions of the near-angle stack and far-angle stack.

EI(7.5o)

EI(22.5o)

Notice the decrease in the elastic impedance value on the farangle stack. 15

EI from logs EI_Near

(a)

EI_Far

(b)

The figures show the (a) crossplot between near and far EI logs, and (b) the zones on the logs. Notice the clear indication of the gas sand (yellow). 16

Gas sand case study

EI at 22.5o

This figure shows a crossplot between EI (7.5o) and EI (22.5o). The background trend is the grey ellipse, and the anomaly is the yellow ellipse. As shown below, the yellow zone corresponds to the known gas sand.

EI at 7.5o

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Scaled Elastic Impedance  One of the problems with EI is the fact that the values do not scale correctly for different angles.  This is due to the variable dimensionality found by raising the velocity and density terms to increasing powers.  Whitcombe (2002), proposed solving this by scaling the EI equation as follows:

 V  EI ( )  VP 0 r 0  P   V P 0  

1 tan 2 

 VS     VS 0 

8 K sin 2 

1 4 K sin 2 

 r     r0 

 ,  

where VP 0 , VS 0 , and r 0 are reference constants.  This also leads us to extended elastic impedance (EEI).  But first we need to review the intercept/gradient method.

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The Intercept/Gradient method  The Intercept/Gradient method is an approach to AVO which involves re-arranging the Aki-Richards equation to:

R P ( )  A  B sin   C sin  tan  , w here : 2

2

2

 VP r  VP  VS r A  R AI   , B  8K  4K , 2V p 2 r 2V p 2VS 2r 2

VS   VP , and K    . C  RVP  2V p VP   This is again a weighted reflectivity equation with weights of a = 1, b = sin2, c = sin2 tan2. 19

The Intercept/Gradient method The amplitudes are extracted at all times, two of which are shown: Offset

+A +B sin2 Time

The Aki-Richards equation predicts a linear relationship between these amplitudes and sin2θ. Regression curves are calculated to give A and B values for each time sample.

-B -A

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The Intercept/Gradient method The result of this calculation is to produce 2 basic attribute volumes

Intercept: A

Gradient: B

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Extended Elastic Impedance  As the next step from scaled EI, Whitcombe et al. (2002) introduced Extended Elastic Impedance, or EEI.  First, they replaced the sin2 term in the two-term AkiRichards equation with tanc, to give the following expression for EEI reflectivity, REEI.

R P ( )  A  B sin 2   R ( c )  A  B tan c  R EEI ( c )  R ( c ) cos c  A cos c  B sin c  Notice that EEI will equal acoustic impedance at c = 0o and gradient impedance (GI) at c = 90o. The limits of c are + and - 90o. 22

Extended Elastic Impedance  This leads to the extended elastic impedance expression, which involves substituting the EEI(c) reflectivity expression into the scaled elastic impedance expression, to give:

 V  p  V  q  r  r  EEI ( )  VP 0 r 0  P   S    ,  VP 0   VS 0   r 0   where p  cos c  sin c , q  8 K sin c , r  cos c  4 K sin c .  Different values of c correlate with various rock properties, as shown on the next slide. 23

Extended Elastic Impedance Vp/Vs EEI(45o)

l

EEI(19o)

K

EEI(10o)

rVs EEI(-45o) m

EEI(-58o)

(a)

(b) Figure (a) shows EEI values at different angles and figure (b) compares elastic parameters to their equivalent EEI curves. Whitcombe et al. (2002)

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EEI inversion EEI(c) Model Volume

EEI(c) Volume Inversion Algorithm EEI(c) Inversion

The EEI inversion approach involves building an EEI(c) model and inverting the EEI(c) volume using an inversion algorithm to create an EEI output.

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Implementation in HRS-9  Now that we have discussed the theory of EEI, let’s see how to implement the process in HRS-9.  The process involves four steps:  Choose a target log and find the optimum c angle.  Build the log parameter model.  Compute the EEI(c) seismic volume from the intercept and gradient.  Perform the inversion.  We have recently built this functionality into HRS-9.  Since the process involves a number of steps, we will first build a Workflow. 26

Creating an EEI Workflow

We start by creating a new workflow Group Name and then start the Workflow Builder. 27

EEI Workflow

The workflow is built by moving processes from the Process to the Workflow list. The final Workflow is on the right.

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Gas sand case study  We will now apply this Workflow to the dataset just described.  As our target log, we have chosen the Vp/Vs ratio.  Note that this will produce the same output as prestack simultaneous inversion.  However, the approach will be different since we are building an EEI model and inverting the rotated intercept and gradient stacks.  In both cases, we are using the pre-stack data as input, rather than the post-stack data.  We will apply both model-based and coloured inversion. 29

Creating A and B volumes First, use the AVO Attribute Volume option to create A and B:

This display is the product of A and B:

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Correlation plot

We then find the maximum correlation value, which is at 39o with a correlation coefficient of close to 1.00. In this case, almost any value between 30o and 60o would work reasonably well. However, in some cases there is a clear peak.

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EEI log spectrum

A good display option is the EEI Spectrum, which shows the EEI computation for every angle between -90o and +90o.

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EEI log curve at c = 39o

Vp

Density

Vs

EEI at 39o

Vp/Vs Ratio

Next, we compute the EEI log at c = 39o. It closely resembles the Vp/Vs ratio log but the units are impedance. 33

EEI model

Next, we compute the EEI reflectivity section, as shown above (Note: REEI(39o) = A*cos(39o) + B*sin(39o)) 34

EEI model The next step is to create the filtered EEI model:

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EEI wavelet

We then extract the statistical wavelet from the EEI reflectivity:

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EEI inversion analysis The results of post-stack inversion analysis are shown here:

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Model-based EEI inversion Model-based inversion with the EEI log is shown here:

The gas sand zone is well defined but the units are impedance. 38

Model-based EEI inversion Here, we scaled to Vp/Vs units using a single scaler:

Now the gas sand zone has the correct low Vp/Vs values.

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Coloured EEI inversion Alternately, we apply coloured inversion, with relative scaling.

Note we still have an excellent definition of the gas sand.

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Conclusions  This presentation has been an overview of the extended elastic impedance (EEI) approach using HRS-9.  I first reviewed post and pre-stack inversion methods.  I then discussed EEI theory and how the general inversion method could be modified to implement EEI inversion.  I then showed how to find an optimum c angle to create the EEI section. I chose Vp/Vs ratio as the target log.  I then showed how to create the EEI section in HRS-9.  Next, I created the inverted Vp/Vs volume, using both model-based and coloured inversion.  Both of the EEI inversion methods gave us excellent definition of the gas sand zone, comparable to pre-stack inversion.

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