Monday, January 18, 2010

Graph theory in action

There is a cool applet on the web that allows you to build and debug pretty complex Graphs. The site is Ashitani's GraphViz and it comes up with a simple example to get you started.

Here is a more substantial example, showing the methodology of estimating permeability from 3D seismic data:

"Cross Plot" [shape=box]
"Measured Data" [shape=diamond]
"Core Porosity" [shape=diamond]
"Permeability" [shape=diamond]
"Logs" [shape=diamond]
"Seismic" [shape=diamond]
"Corrected Porosity" [shape=box]
"Seismic Porosity" [shape=box]
"Seismic Perm" [shape=box]
"k-phi" [shape=box]
"Neutron,DT,RHO"->"Log Porosity"
Core->"Core Porosity"
"Core Porosity"->"Corrected Porosity"
"Log Porosity"->"Corrected Porosity"
Impedance->"Seismic Porosity"
"Corrected Porosity"->"Seismic Porosity"
"Seismic Porosity"->"Seismic Perm"
"k-phi"->"Seismic Perm"
"Corrected Porosity"->"k-phi"

These lines describe relationships shown in the figure below. To make your own version of the figure, copy the lines, paste into Ashitani's input window, and hit the return key. The graph image can be saved in the usual way with right click and Save As...

Anyone who has struggled with PowerPoint or a drawing program to make such graphs will appreciate this cool tool. Behind this applet is the GraphViz language which has many options like arrow types, colors, fonts, orientation, etc. If a reader who knows more than I sends a great example using extended functionality, I will let you know.

Sunday, January 10, 2010

The Meaning of Seismic Amplitude

[Note: A version of this blog entry will appear in World Oil (Feb, 2010)]

In this column we consider those reservoir properties that affect amplitude as observed on seismic data, but first we need to acknowledge the many amplitude effects associated with seismic energy passing through the overburden (that section of the earth between the acquisition surface and the reflection interface). We can imagine riding along with a seismic ray as it progresses from source to receiver, and a wild ride it would be. The source has a radiation pattern, meaning that waves going straight down will be of a different strength than those at other take-off angles. As the ray travels through the earth, it experiences continuous amplitude loss due to geometric spreading, and further continuous loss due to absorption. Each time the ray passes through an impedance contrast interface the amplitude is scaled by a transmission coefficient (which is different for upgoing and downgoing waves) leading to cumulative transmission losses. In the reflection process the amplitude is scaled by an angular reflection coefficient whose mathematical form depends on the kinds of media in contact. Progressing back up toward the acquisition surface, our seismic ray amplitude is further reduced by more geometric spreading, more absorption, and more transmission loss. Finally, the amplitude measured by the receiver is a function of the receiver coupling and directional sensitivity.

In seismic data interpretation, it is tacitly assumed that all amplitude effects described above are compensated for by data processing and migration, leaving reflection events with peak amplitude proportional to reflection coefficients. For prestack seismic data, the angular elastic P-P reflection coefficient, Rpp, is assumed to be embedded in the data. Based on the Zoeppritz equations, Rpp is the basis for amplitude versus offset (AVO) interpretation. For migrated stack data, the simpler normal incidence reflection coefficient, R0, forms the foundation of amplitude interpretation. The critical assumption in poststack interpretation being that observed peak amplitude variations are proportional to underlying R0 changes. But how accurate is this assumption?

Real seismic data is shot with a range of offsets and collapsed back to zero offset through CMP stacking. If we consider an isolated, isotropic, elastic interface in the earth, its reflection amplitude as a function of incidence angle can be approximated with a simple function. Modern marine shooting will give an incidence angle range of about 0-30 degrees at reservoir levels. The seismic stacking process acts like a summation over offset of all amplitudes in this range, something that we can simulate mathematically. The result for a typical Gulf of Mexico gas sand reflection shows that only about 50% of the stack amplitude can be attributed to R0, the rest being due to other factors. So the assumption that stack amplitude is proportional to R0 has a large error bar in this case, although a gas sand has extreme properties. Even so this result can be taken as a cautionary tale.

With all these amplitude caveats understood we can proceed to consider the poststack case and ask the following question: What are the reservoir property changes that best explain seismic amplitude variation observed along horizons in poststack 3D seismic data?

One approach to this problem is reservoir property prediction using seismic attributes. For example, hydrocarbon pore thickness (HPT; net pay times porosity times hydrocarbon saturation) can be posted at well locations and compared to an attribute map. If a cross plot shows promising correlation, we can use it to predict HPT between wells from the attribute. A problem with this approach is that a small number of wells and a large number of candidate attributes can result in a misleading or unreliable estimator.

Another approach is to use a physics-based rock model describing the reservoir as a porous fluid-saturated rock. We begin with parameters describing environment, fluid, and frame from well log or laboratory measurements. The theory also involves one or more free parameters used to localize the model and make it specific to the reservoir under study. Log or core measurements of seismic P-wave velocity are adjusted using an appropriate upscaling theory and used for calibration. Further calibration is often is based on a crossplot of log/core velocity and porosity. Regression on these data points allow determination of the free parameters and the result is a predictive rock model that can be used to compute velocity, density, impedance, interval transit time, and reflection coefficients in response to changes in frame, fluid, or environment parameters. Along with our assumption that seismic amplitude and reflection coefficients are proportional, the rock model forms the core of quantitative seismic interpretation.

It is usual to take the physical basis of amplitude variation in response to rock property changes to be the Gassmann theory, a 1950's model well-suited to sandstones. The nature and mathematical structure of rock models for carbonate, tight sandstone, and shale are current areas of research. The predictive rock model is invoked to explain amplitude changes of various kinds in seismic data. These include lateral changes in a 3D survey mapped along a particular horizon, or AVO effects, or even time-lapse phenomena associated with multiple vintages of 3D seismic data during reservoir production (or injection). These associations are the foundation of amplitude interpretation, attribute analysis, and reservoir property prediction.

Can seismic data be interpreted without a calibrated rock model? Certainly. Structure can be completely characterized, stratigraphic elements based on reflector geometry and robust amplitude effects can be mapped. But the rock model becomes supremely important when interpretation is pressed into the regime of reservoir property prediction.

As a predictive rock model example, consider the Glenn Sandstone in the Glennpool field of Oklahoma. This is a shallow, giant oilfield discovered 1905. Based on the Gassmann theory and modern digital well logs, it is possible to make a predictive rock model. The details would take us too far afield, but the conclusions are brief and interesting.

If a modern, 3D seismic survey were to be shot over the Glennpool field (one was in 2008), and the Glenn Sandstone event mapped, we would see lateral changes in amplitude. How are these to be explained in terms of reservoir properties? We will consider a benchmark R0 based on brine saturated average rock properties known for the Glenn Sandstone.

Gas is a strong effect (220% R0 boost relative to the benchmark), but there is no evidence of a gas cap or even significant gas-oil-ratio in this field, so we eliminate gas as a candidate. Porosity ranges from 15-30%, with an average of 24%. The effect on R0 is dramatic. At 15% porosity the R0 change is -100% (impedance-matched situation), while at 30% porosity we get a 60% R0 boost. So we hold on to this possibility. We can also predict the effect of clay fraction in the Glenn Sandstone, something that has a clear relation to permeability. A change of 0 to 20% clay fraction increases R0 by about 6%. Since clay content is known to vary in the Glenn, this effect is a valid candidate. Finally, after primary, secondary, and tertiary oil production, the residual oil saturation is still over 70% in the Glennpool field. As oil saturation drops from 1.0 to 0.7, the R0 decline is about 3.5%. This is a small, but potentially mappable, effect.

In summary, the most likely reservoir properties that can be predicted by seismic amplitude for the Glenn sandstone are porosity, clay content, and oil saturation (in that order). Knowing this, it is highly probable that a suitable amplitude-based seismic attribute would correlate strongly to a composite reservoir parameter formed by combining these properties.

This exercise with the Glenn sandstone cannot provide general results applicable to other reservoirs. The Glenn is a Pennsylvanian-age midcontinent sandstone, representative of stiff, high velocity sandstones. In a stiff formation like the Glenn, pore fluid changes do not influence impedance of the rock nearly as much as they would in rocks from a soft-rock basin like the Gulf of Mexico. But our discussion does illustrate the procedure for linking reservoir rock properties to seismic quantities in poststack 3D seismic data.

Elements of 3D Seismology (2004, Chapters 6 and 26).

Figure 1. The Concept of a predictive rock model

Figure 2. Predicted R0 effects for the Glenn Sandstone calibrated rock model.

Tuesday, January 5, 2010

Wolf Ramp

I've been putting the final touches on a paper with Bernard Bodmann on seismic reflection from a vertical velocity transition zone. The original paper by Alfred Wolf is in 1937 Geophysics. It turns out that for vertical incidence an exact result can be found (Wolf did it), an we were interested in looking at the solution in time-frequency space. Anyway, after much labor the paper is almost done, should be ready by end of the week.

Next up are papers on the dynamic Fresnel zone, a graphical tour of Zoeppritz sensitivity, and maybe a few others. But before that, I will be starting a class this Friday afternoon and continuing Saturday, and so on for 3 weeks (a professional MS course on 3D seismic interpretation). And the regular semester starts Jan 19....