When Gassmann Fails

Four Assumptions, Four Ways to Fail

Gassmann earned four assumptions in Part 4.1, and each one, read as a hazard rather than a premise, names a way the theory breaks. This is not a catalogue of a fragile method. Gassmann is excellent inside its domain, and that domain is wide. But the domain has edges, and knowing where they lie is the whole difference between a substitution you can defend and one that will quietly mislead.

Frequency, Drainage, Distribution, Direction

High frequency. Gassmann is the relaxed, low-frequency limit (Part 0.4): it assumes pore pressure equilibrates through the rock within a single wave period. At sonic-log frequencies, and much more so at ultrasonic laboratory frequencies, it cannot. The fluid is trapped where it sits, the rock responds unrelaxed, and it is stiffer than Gassmann predicts. A substitution calibrated on ultrasonic core measurements therefore over-predicts the saturated stiffness when it is carried to the seismic band. Gassmann belongs at seismic frequencies; moving a lab number in, or a Gassmann prediction out to the lab, without a dispersion correction is the category error Part 0.4 warned against.

Shales and tight rocks. Even at seismic frequency, pressure equalizes only if the fluid can move. In a shale or a very-low-permeability rock the fluid cannot drain within a wave period, so the relaxed condition fails from the drainage side rather than the frequency side. Worse, clay and water interact chemically, violating the no-softening assumption outright. Gassmann is a clean-sand theory; on shales it is a rough guide at best. Patchy saturation. Gassmann wants one effective fluid filling the pores uniformly. When gas and brine sit in separate patches (Part 3.6), there is no single effective fluid: the correct move is to saturate each patch with Gassmann on its own fluid and then mix the two saturated rocks, which gives the stiffer patchy answer, not the soft result a single Gassmann pass on the Wood mixture would produce. Anisotropy and mixed mineralogy. Gassmann assumes an isotropic, ideally single-mineral rock; strongly laminated or fractured rocks are anisotropic, and rocks built from several very different minerals strain the single mineral-modulus picture. Both call for the anisotropic Brown-Korringa generalization or a carefully chosen effective mineral.

The Rule

Read the figure's verdict as the working rule. Gassmann is excellent for a clean, connected, isotropic sand undergoing a water-to-hydrocarbon substitution in the seismic band, and it needs care, a correction, or a different tool everywhere else. That is not a narrow license: most fluid substitution done in practice sits squarely inside it. But the four edges are real, and each has produced a wrong answer for someone who forgot it was there. The discipline is to check all four before trusting a result, not after being surprised by one.

With the theory (Part 4.1), the workflow (4.2), the anatomy (4.3), the error budget (4.4), and now the caveats all in hand, one honest thing remains: run the whole of it on a real well and produce the deliverable an interpreter is actually paid for. The last section of the part substitutes gas, and oil, into a brine sand on the Ogbon-1 well and reads the before-and-after logs, all the way to the reflection at the top of the sand.

References

• Mavko, G., Mukerji, T., & Dvorkin, J. (2009). The Rock Physics Handbook (2nd ed.). Cambridge University Press.
• Wang, Z. (2001). Fundamentals of seismic rock physics. Geophysics, 66(2), 398-412.