Fluid Contacts from Pressure

Chapter 15, Formation Pressures and Fluid Contacts

Learning objectives

Find the depth where two fitted pressure gradients cross, and say what it is: the free-water level, not the contact the logs see
See that no pressure point need sit exactly at the crossing
Relate the crossing's reliability to the gradient contrast, and compute its standard error from the two fits
Recognize a gas-water crossing as crisper than an oil-water one

Two Gradients, One Crossing

Section 15.2 turned the slope of one fluid's pressures into its density. A column with two fluids in it gives two lines on a plot of pressure against depth, the oil above and the water below. Pressure rises faster with depth in the water, 0.455 psi/ft against 0.347 psi/ft in Ogbon-1's oil, so the lines approach each other going down and meet at one depth, where the oil and the water are at the same pressure. Fit a straight line to the tests in each fluid, $p = p_0 + g\,z$ with the pressure at the surface $p_0$ and the gradient $g$ , and the crossing is the depth where the two fits are equal:

z_c = \frac{p_{0,o} - p_{0,w}}{g_w - g_o} .

No test has to sit at that depth, and in Figure 15.3 none does. The ten tests are the figure's own teaching survey, not the well's survey of Section 15.6: it spreads its tests up the column and lets the fluid over the water change, which the well's cannot, and its default gauge noise, 1.0 psi, is four times the well survey's 0.25 psi, so that the crossing's uncertainty is large enough to see. Its plan: five in the oil, from 5880 ft to 6140 ft, and five in the water, 8 ft to 48 ft below the free-water level, each the true pressure plus gauge noise. The nearest test is 8 ft from the free-water level, and the oil line is extended 110 ft below its last test to reach it. Each line is solid across its tests and dashed beyond them, because beyond them the line is an extrapolation.

What the Crossing Is

At the crossing the two pressures are equal, so the capillary pressure is zero. That depth is the free-water level, as Section 11.3 defines it, and it is not the oil-water contact, the depth below which the rock holds only water, where the logs first see oil. The contact stands one entry height above the level, because oil does not enter the rock until its buoyancy beats the rock's entry pressure: in Ogbon-1, 4.0 psi over 0.108 psi/ft, 36.9 ft, so the contact is at 6213.1 ft and the level at 6250 ft. Take the gauge noise away and the lines cross at 6250.0 ft exactly, 36.9 ft below the contact the logs see. A pressure survey that sets out to find the contact finds the level. Smolen notes that a formation tester's pressure profile can locate the gas-oil and water-oil contacts; the book keeps its two words apart, so what the profile locates is the level.

The figure shows one draw of 1.0 psi of noise, a fixed seed chosen as a typical draw (another lands elsewhere, as the 400 re-draws below show). In it the lines cross at 6262.6 ft, 12.6 ft below the level and 49.5 ft below the contact. A colleague who sees the 49.5 ft and says that the pressures and the logs disagree is mixing two things: 36.9 ft of it is the entry height, which is physics and not an error, and the 12.6 ft left over is the survey's own miss, 0.8 of its standard error, 15.9 ft. Section 15.4 takes the free-water level as its subject.

How Well It Is Pinned

A fitted line is known best at the middle of its tests and worse away from them. With gauge noise $\sigma$ and $n$ tests at depths with mean $\bar z$ and spread $S = \sum (z - \bar z)^2$ , the ordinary least-squares line has a variance $\sigma^2\,(1/n + (z - \bar z)^2/S)$ at depth $z$ . The two lines differ by the gradient contrast $g_w - g_o$ for every foot, so to first order an error of one psi in their difference moves the crossing by one over the contrast, and the standard error of the crossing is

\sigma_z = \frac{\sigma \sqrt{1/n_o + (z_c - \bar z_o)^2/S_o + 1/n_w + (z_c - \bar z_w)^2/S_w}}{\left| g_w - g_o \right|} .

Proett and Storm (2008) describe a statistical analysis of this kind: the intersection of the fitted gradients is taken as the fluid contact (the depth this book calls the free-water level), and a Monte Carlo simulation, which assumes linear regression and varies the pressures and depths by their expected standard deviations, gives its uncertainty. In the default survey the numerator, the uncertainty of the difference between the two lines at the crossing, is 1.7 psi, and over the contrast of 0.108 psi/ft it is 15.9 ft. It grows in proportion to the noise: at 2.0 psi it is 31.8 ft. The formula is a first-order result, and the figure checks it against 400 fresh draws of the noise, each fitted and crossed again: the middle two thirds of them spread over 16.9 ft either side of their median, close to the formula but not equal to it, because the crossing is a ratio of two noisy quantities.

The contrast is the other lever. A heavy oil of 0.90 g/cc, the heaviest oil of Figure 1.5, has a gradient of 0.390 psi/ft, close to the water's 0.455 psi/ft, so the lines cross at a shallow angle. The contrast falls to 0.065 psi/ft and the same 1.0 psi of noise costs 26.5 ft instead of 15.9 ft. A heavy oil also lifts the contact higher above the level, to 61.5 ft above it, at 6188.5 ft. At 2.0 psi of noise the standard error is 52.9 ft and the level and the contact are 1.2 standard errors apart: a survey this poor cannot say which of the two its crossing is. The formula also says what would help: a better gauge, more tests, or tests spread wider and nearer the crossing.

Gas Is Crisper Than Oil

Swap the oil for a gas and the same ten tests do much better. The figure takes the Rock Physics kernel's Batzle and Wang gas of gravity 0.65 at 50 °C, at the pressure of the free-water level, 0.166 g/cc, which is a gradient of 0.072 psi/ft: a hypothetical gas, since Ogbon-1 has none. The contrast with the water rises from 0.108 psi/ft to 0.383 psi/ft, 3.5 times as much, and the standard error falls from 15.9 ft to 4.5 ft, 3.5 times less. The uncertainty of the lines, 1.7 psi, has not changed. Only the number it is divided by has, and that is why a gas-water crossing is crisper than an oil-water one. The gas line is taken as straight: its density changes by 0.8 % up to the shallowest test, which would bend it by 0.1 psi, well inside the noise. The figure does not draw a gas-water contact the logs would see, because the book has no gas-water entry pressure; its entry pressure is an oil-water one.

What the Figure Leaves Out

The noise is independent and Gaussian, one fixed draw of it, which is the best a gauge can be. Real surveys add errors that are not random in that way, such as a pressure that has not yet settled (Section 15.1) or a gauge that reads high. The lines are straight and the water is hydrostatic from the surface. And the method assumes that one fluid system connects every test: tests in separate compartments (Section 15.5) lie on lines that need not cross at any common level.

References

Proett, M. A. and Storm, B. H., Jr. (2008). Method and system for statistical pressure gradient and fluid contact analysis. US Patent 7,444,242 B2.
Smolen, J. J. Wireline formation testers. In Development Geology Reference Manual, AAPG Methods in Exploration. AAPG Wiki, wiki.aapg.org/Wireline_formation_testers.