Velocity Is a Model

Part 0: The Bridge

Learning objectives

  • Predict the same sand's velocity with four defensible models and watch them disagree
  • State what kind of claim each model makes: heuristic average, empirical revision, lab regression, frame physics
  • Explain why no lookup table can settle the disagreement
  • Use the spread between models as an honest uncertainty rather than an embarrassment

The Lookup That Fails

If velocity were a rock property in the way density of quartz is, there would be a chart: porosity in, velocity out, one number. The literature instead offers a family of respected answers. The Wyllie time average treats the rock as fluid and mineral travel times in series, dfrac1VP=dfrac1−phiVma+dfracphiVfl\dfrac{1}{V_P} = \dfrac{1-\phi}{V_{ma}} + \dfrac{\phi}{V_{fl}}P=dfrac1−phiVma+dfracphiVfl. Raymer-Hunt-Gardner revised it from log experience, VP=(1−phi)2Vma+phi,VflV_P = (1-\phi)^2 V_{ma} + \phi\, V_{fl}P=(1−phi)2Vma+phi,Vfl. Han fitted laboratory sandstones at 40 MPa and reported a plane, VP=5.59−6.93,phi−2.18,CV_P = 5.59 - 6.93\,\phi - 2.18\,CP=5.59−6.93,phi−2.18,C with CC the clay fraction. And a critical-porosity frame model builds the dry rock first, softening the mineral toward the critical porosity, then saturates it with Gassmann. Four pedigrees: a heuristic, an empirical revision, a straight regression, and a piece of frame physics.

A Fair Contest

The figure runs all four on the same rock: the same quartz-clay mineral mix, the same brine, the same porosity axis. Nothing is rigged.

Four models, one rockporosityVpthe spreadFour published models, one rock: the disagreement is the honest size of what the models are not told, the texture.

Park the marker at twenty-five percent porosity in a clean sand and read the damage: the predictions sit more than half a kilometer per second apart, roughly the difference between a gas sand and a brine sand in many basins, and every one of the four is used, published, and defensible. Slide the clay up and they disagree about that too. None of them is wrong; each answers a slightly different question. Wyllie describes consolidated, cemented rock at moderate porosity. Raymer repaired its known bias. Han reports what seventy-odd real sandstones did in a press. The critical-porosity model is the only one that even tries to describe the frame, and it is the crudest of its family.

Disagreement Is Information

The spread is not noise to be averaged away; it is the honest size of what you do not yet know about the rock: its texture, its cement, its stress state. That is why this course does not begin by handing you a favorite equation. It begins, in Part 2, with bounds: limits that hold for every possible arrangement of the same ingredients, so that every model must live inside them, and a measurement outside them means something is wrong: the number, or the composition it was computed for. Before that, one more piece of motivation: the next section takes the chain one step further and asks what a seismic reflection is actually made of.

References

  • Mavko, G., Mukerji, T., & Dvorkin, J. (2009). The Rock Physics Handbook (2nd ed.). Cambridge University Press.
  • Wyllie, M. R. J., Gregory, A. R., & Gardner, L. W. (1956). Elastic wave velocities in heterogeneous and porous media. Geophysics, 21(1), 41-70.
  • Raymer, L. L., Hunt, E. R., & Gardner, J. S. (1980). An improved sonic transit time-to-porosity transform. SPWLA 21st Annual Logging Symposium, Paper P.
  • Han, D.-H., Nur, A., & Morgan, D. (1986). Effects of porosity and clay content on wave velocities in sandstones. Geophysics, 51(11), 2093-2107.

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