Thursday, May 16, 2013

De Rerum Fractura (On the Nature of Fractures)

Update 5/16/2013:
1. Better form of equation in original post
2. First look at Weibull fracture model for the Mississippian Boone limestone (and chert) based on data in Buckland (2013).   This addresses the question: How much stress (pressure) will it take to get an N% fracture efficiency in the Miss Lime?  The experiments were done at ambient pressure, thus representing unconfined fracture stress. So the answer is: At room conditions, application of 3150 psi will fracture about 65% of the samples (representing all major facies in the Boone; LS, hard chert, tripolite, and mixtures thereof). We are working to scale this data to reservoir conditions as an aid in designing frac jobs in the Miss Lime oil play of Oklahoma and Kansas.

Figure: Unconfined fracture probability for the Boone. 

Reference:
Buckland, K., 2013, A Geomechanical Study of the Mississippian Boone Formation, MS Thesis, Department of Geology, University of Arkansas [available now to MArkUP sponsors]


**** Original post ****

[Note: A version of this blog entry appears in The Leading Edge (October, 2010)]

“…choose a simple function, test it empirically, and stick to it as long as none better has been found.”
Waloddi Weibull, Swedish scientist (1887-1979)

These days it is much in the mind of petroleum seismic researchers to get into the “fracture game.” This is driven by the remarkable rise in natural gas reserves from shale reservoirs, primarily in the US for now but quickly being exported worldwide. Shale formations have long been of interest as source rock and reservoir seal, but in the last 20 years shale has taken center stage as a reservoir in its own right. Gone are the days when we would drill through shale and get a gas kick, ignore it, then drill on down to conventional reservoirs. Gas shale is considered an unconventional reservoir, mainly because of very low porosity and permeability compared to those Swiss cheese sandstones and limestones. For a 25% porosity conventional reservoir, you need only to punch a hole through it, open the choke, and production will scream out for decades. But shale, like tight gas sand, must be modified to open channels for production. Wells must be horizontally drilled to expose more reservoir rock and they must be fractured.

This brings up the subject of fracturing rock. Many of us are fairly well versed in seismic wave propagation, elasticity, and rock properties, but what about crack nucleation, stress at a crack tip, or failure criteria? I seem to recall a structural geology course long ago that talked about Mohr’s envelope, fracture orientations relative to regional stresses, hysterysis loops, and vast amounts of similarly confusing topics. Pictures were shown of real rock fractures, but theory was set in simple media, typically isotropic elastic. Fractures in the real world, like earthquakes, seemed easy to describe and hard to predict.

If we look beyond rocks, there is a rich literature and body of knowledge on fracturing of ceramics, glass, and other manufactured materials. It is interesting to see how they approach fracture theory and prediction. Because the materials are man-made, their purity of structure can be engineered and measured. It soon becomes clear that fracture tendency is related not so much to the strength of the ideal material, but the density and distribution of flaws. Fracturing is a process that begins at the molecular level, not where the 3D architecture of the material marches on uniformly, but at flaws. These flaws amplify stress and can initiate rupture, but it is anyone’s guess which flaw will be the culprit. It is sometimes useful to abstract a problem to a more general, simpler one.

The lead quote by Waloddi Weibull comes from a 1951 classic, A Statistical Distribution Function of Wide Applicability. When Weibull says “wide applicability” he means it; the paper has examples that include yield strength and fatigue life of steel, particle size distribution in ash, cotton fiber strength, height of adult males in the UK, and width of beans.

The concept at the heart of Weibull’s approach is, literally, that of the weakest link. Imagine a chain composed of several links, and we want to predict when it will fail as we pull on it. We are thinking about a real chain here, not some kind of theoretical construction with particular link geometry and ideal material properties. We need to know if we pull on a real chain with a certain tension, can we depend on it?  How likely is it to fail?

Note the statistical nature of the problem. We immediately give up a deterministic worldview for this calculation. Who would trust a method that came back with a yes or no answer? The answer, like the problem, is inherently statistical. The approach to this problem is part experimental and part theoretical. From the manufacturer, we get a big box of individual chain links. The experiment involves lab testing each link in the box at a certain stress. We set up 10 stress levels to test. At each stress level we pull on 100 individual links, each time noting if it failed or not. Altogether, then, we have tested 1000 links.

Now analyze the data a little bit. At each stress level, calculate the fraction of tested links that failed, representing the probability of failure at this stress. If we plot the data with horizontal axis being stress and vertical axis being probability of failure, it will look something like the dots in Figure 1.

Figure 1. The Weibull function can be used to characterize failure probability based on experimental results, yielding a predictive fracture model.


The last step is the theory part. In the language of statistics, our “probability of failure” is a cumulative distribution function, or CDF. What Weibull figured out from thinking about chain links is this CDF can be represented by the function


where s is stress in mega-pascals (MPa), m is the dimensionless Weibull modulus, sigma _sub_zero is the critical stress in MPa, and P(s) is the probability of failure at a given stress. The critical stress is defined as the stress at which the failure rate is 63%.  If this function seems too simple to be true, reread the lead quote. The points in Figure 1 have been fit with a Weibull CDF curve (purple line) and the fit parameters are shown in the plot title.

After doing all this work, we are ready to answer a question about the chain. If I tug on this chain with a big elephant (140 MPa), how likely is it to break? Answer, about 30% chance it will break. You should be ok, unless the elephant gets mad.

What does this have to do with fracturing shale? Everything. The scenario we just described is exactly on point.  Here is the plan.  Core a few hundred feet of your favorite shale.  From this take 1000 core plugs. Decide on the stress levels you want to test. In the lab, put the core plugs one at a time in a machine that ramps up to one of the programmed stresses.  Note if it fails or not.  The rest of the exercise is just like the previous example.

It is a lot of work, and not cheap, but the result will be a predictive model of rock fracture probability. Shale gas wells cost US $5–10 million and the Weibull parameters can say when fractures are likely to initiate and how far they are likely to extend (stress drops away from the well bore).

In the SEG technical literature archive are thousands of papers; the annual meeting alone generates more than 800 expanded abstracts. Lots of work on fractures, monitoring fractures, mapping fractures, fracture mechanics, and so on. Now here is something really interesting. A full-text search on all SEG publications for the term “Weibull” turns up a simple result: Zero (out of 26,950 items published in Geophysics, Leading Edge, and SEG Abstracts as of Oct 11, 2010).

Suggested reading. A Weibull statistics example by Megan Frary (Boise State University) can be found at the Wolfram Demonstrations Project. The plot in this article is based on this demonstration. Weibull’s 1951 paper can be found in PDF format linked to Dr. Robert Abernathy's Weibull Page.

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Dear Professor Liner:

Your piece in Seismos, TLE October, 2010 really sharpened my memory and took me way back to the early fifties. I enjoyed the article very much but I am surprised that Waloddi Weibull is not mentioned in an issue of GEOPHYSICS of that time period.

At that time I was working at The Atlantic Refining Company’s Geophysical Laboratory in Dallas. Dr. J. P Woods, the director held technical sessions on a regular basses where we discussed new or theoretical subjects. Maybe Dr. Woods found his writings somewhere, but I can remember Weibull’s ideas of predicting failure having been a subject at a couple of our afternoon meetings.

Although we spent time working with geophysics, elastic wave theory, etc. our main work at the lab was engineering. We were primarily concerned with the design and production of seismic instruments (analog, vacuum tube). Weibull may have come to us through other publications such as that of the IEEE. Early fifties seem a little early for failure mode analysis but it became very important in the late fifties and early sixties. I do remember things being written about the problems with the very large vacuum tube computers…dealing with tube failure.

Professor Liner, I hope I haven’t board you with my reminiscing but I appreciate anything that stirs up the old gray matter.

The footnote to your Column said you read old books. Just how old and what kind? Are you a book collector? As I approach my terminus I look for a home for some of my “old stuff”.

Regards

Morris Gillett
Birchwood, WI

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Morris,

Thank you for the interesting letter. I suspect there are many out there working with Weibull modulus. One interesting aspect is that the concept is quite general and you can get different Weibull moduli depending on the kind of stress applied in the lab. For example, you could do extension, compression, shear, etc and each would have an associated Weibull modulus. I suppose the ideal experiment for, say, the Eagleford shale gas play, would be to prestress the shale core to in situ conditions (P and T) then simulate injection of cold high pressure fluid to actually frac the rock. In this way, one would develop a Weibull modulus describing the probability of fracture development as a function of injection pressure for that shale at
depth. Then we could answer the question, what is the Weibull modulus for the Eagleford? Surely this would be an important aid in planning frac jobs.

As for the old books, I have a good collection of applied seismology texts (1930-now), early modern physics (1880-1960), and then various topics in English antiquarian books (1580-1750) including mathematics, physics, archeology, and history. English translation of Greek and Roman classics are a favorite.

Best regards,

Chris Liner

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