End of the Rainbow

[Note: A version of this blog entry appears in World Oil (December, 2010)]

Over the last 12 months this column has covered topics from shale gas to seismic migration. When it comes to seismic data used for hydrocarbon exploration and production, the pot of gold, so to speak, is the Reflection Coefficient (RC) arising from layer boundaries deep in the earth. The RC indirectly contains information about rock and fluid properties. I say indirectly, because the RC mathematically just depends on velocity and density, but these in turn depend on porosity, pressure, oil saturation, and other reservoir parameters. One key parameter the RC does not even claim to supply is permeability, although it is sometimes estimated using an elaborate workflow involving well logs, core, and seismic (See this blog entry). Another, more direct, seismic permeability estimator is receiving attention these days and that is what I would like to talk about.

We won't be writing down any equations, but reflection coefficients are mathematical in origin. The simplest case involves a seismic wave traveling perpendicular to a layer boundary in an earth characterized by velocity and density. When this wave hits the boundary it splits into reflected and transmitted parts. The size of each is determined by two Boundary Conditions (BC) at the interface, conservation of energy and continuity of amplitude. The two BCs lead to two equations in two unknowns, the reflection and transmission coefficients. Because it is based on a simple model of the earth where each point is described by P-wave speed and density, the RC found in this way depends only on these parameters. Importantly, the RC in this case is just a number, like 0.12, and it is the same number if a 10 Hz wave is involved or 100 Hz.

Now imagine a more complicated view of rock involving such things as porosity, mineralogy, permeability, and pore fluid properties (modulus, density, saturations, viscosity). A theory of waves traveling through such a porous, fluid-saturated rock was developed in the 1950s by Maurice Biot. It is the foundation of poro-elasticity theory and the subject of hundreds of papers making readers worldwide thankful he had a short name. Some odd predictions came from the Biot theory, in particular a new kind of P-wave. The usual kind of P-wave (called fast P) is a disturbance traveling through the mineral frame of the rock, but influenced by the pore fluid. New wave (slow P) is a sound wave in the pore fluid, but influenced by the rock frame. In particular, the slow wave has to twist and turn through the pore space compressing and decompressing fluid and thus has a natural connection to permeability. Before long the slow wave was seen in the lab and the theory set on firm experimental footing.

By the 1960s, researchers figured out how to calculate the reflection coefficient for an interface separating two Biot layers. Since there are three waves in each layer (fast P, slow P, and S) there are 3 reflected and 3 transmitted wave types, meaning we need 6 boundary conditions to solve everything. I won't rattle them off, but there are indeed 6 BCs and the reflection coefficient was duly found, although it is enormously complicated. It would take several pages of small type equations to write it down. As you can imagine, it took a while for people to understand the Biot Reflection Coefficient (BRC), a process by no means completed.

One tantalizing feature of the BRC is its dependence on permeability and pore fluid viscosity. This holds the hope of mapping things of direct use by reservoir engineers, and doing it without punching a hole in the ground. But things are not as easy as that. These important properties are competing with porosity, mineralogy, and other rock properties to influence the BRC. If the BRC were just a number, like the classic RC, then there would be little hope of unraveling all this.

But the BRC is not just a number, it is dispersive (a function of frequency). This means that a low frequency wave will see a different BRC than a high frequency one. It may not seem like this is much help, but there has luckily been a decade or two of research and development on something called Spectral Decomposition (SD). Like white sunlight bent and split by water droplets to form a rainbow of colors, SD pulls apart a broadband seismic trace into its constituent frequencies. This fancy trick has lead to a universe of seismic attributes revealing ever more geological detail in 3D seismic data.

One result of SD applied around the world is a growing realization that seismic data is always a strong function of frequency. We shoot seismic data with a bandwidth of about 10-100 Hz, but looking at, say, the 20 Hz part we see quite a different picture than 30 Hz, or 40 Hz. The main reason for this is a complex interference pattern set up by classical reflection coefficients in the earth. But researchers and companies are also thinking about mining this behavior for the frequency-dependent Biot reflection coefficient.

The BRC is naturally suited to high-porosity conventional sandstone reservoirs. But shale also has some very interesting properties that may be illuminated by the BRC. We now understand that a vast spectrum of rock type goes by the name of 'shale'. These rocks tend to have low (but variable) permeability, and anomalous attenuation affected by fluid viscosity that is dramatically different for gas, condensate, and oil.

There is much work to do in following this rainbow, but unraveling the many competing effects is a next logical step in seismic reservoir characterization. Stay tuned for Biot Attributes.

A fond farewell... With this column my year as a World Oil columnist comes to an end. Other duties call, including a book project titled A Practical Guide to Seismic Dispersion, requiring my full attention for the next few months. I have the deepest appreciation for the WO editorial staff who gave me this exceptional opportunity, and to the many readers who wrote with their thoughts on the column. You can keep up with me, as always, through my Seismos blog. Adios.