The Vertical Stress from a Density Log

Part 1, Part 1: Stress, the Tensor and the Circle

Learning objectives

Compute the vertical stress as the integral of density times gravity down the column
Run the integration on a density log and read Sv, its gradient, and its psi/ft equivalent at any depth
Quantify how an error in the shallow, rarely-logged section propagates all the way down the Sv profile
State the offshore corrections: sea water contributes its own weight and the air gap contributes none

The One Stress You Can Just Compute

Of the three principal stresses the earth hands us, two will take this course five more parts to pin down. The third you can compute before lunch on day one: the vertical stress is nothing but the accumulated weight of everything overhead, $S_v(z) = \int_0^z \rho(z')\,g\,dz'$ rho(z′),g,dz′. Feed it the bulk-density log, sum trapezoid by trapezoid, and the answer is as good as the log. No model, no assumption about tectonics, no calibration: arithmetic. That is why every stress study starts here, and why $S_v$ v serves as the reference against which the horizontal stresses are hunted in Parts 5 and 8.

Integrate the log in the figure. With the density profile compacting from about 2.05 g/cc at surface toward 2.55 at depth, averaging the familiar 2.3, the running total reaches 67.7 MPa at 3000 m: a gradient of 22.6 MPa/km, one point zero psi/ft, the canon overburden that every later part leans on. Then abuse the shallow section with the slider. Density logs are rarely run in the first few hundred meters, where holes are washed out and tools disagree, and an error there is not local: it is a constant offset carried by every depth below it. A tenth of a gram per cubic centimeter across five hundred meters is half a megapascal forever. Fill the gap deliberately, from check-shots, regional compaction trends, or nearby wells, and say in the report that you did.

Offshore, and the Honest Datum

Two bookkeeping rules finish the job. Offshore, the water column pushes too: start the integral with the weight of sea water, about 10.1 MPa per kilometer of water depth, before the sediment begins. And the air gap between the rig floor and the sea adds nothing: air is weightless in these units, and confusing drilling datums with the seafloor or sea level is a classic way to smear half a megapascal of error across a stress study. State the datum, integrate honestly, and $S_v$ v becomes the firmest number in the whole model, which is exactly what Part 8 will treat it as when the Ogbon-1 ledger is assembled. Next, Part 2 asks what the rock does under these loads: strain, stiffness, and the poroelastic bookkeeping between them.

References

Zoback, M. D. (2007). Reservoir Geomechanics. Cambridge University Press.
Fjaer, E., Holt, R. M., Horsrud, P., Raaen, A. M., & Risnes, R. (2008). Petroleum Related Rock Mechanics (2nd ed.). Elsevier.
Jaeger, J. C., Cook, N. G. W., & Zimmerman, R. W. (2007). Fundamentals of Rock Mechanics (4th ed.). Blackwell.