Reading inversion products: elastic to rock properties

What a simultaneous inversion delivers, and how well

Written in the contrasts of the logarithms of the three properties, each half the jump across an interface, the linearised Aki–Richards equation of Section 5.4 becomes (Fatti et al. 1994)

with $k = V_{\mathrm{S}}/V_{\mathrm{P}}$ of the background. The three coefficients decide what an inversion can recover:

• P-impedance has a coefficient near 1 at every angle, so every stack measures it. It is the best-constrained product.
• S-impedance enters through $8k^2\sin^2\theta$, which is zero at normal incidence and grows with angle, so it needs the mid and far stacks.
• Density enters through $\tan^2\theta - 4k^2\sin^2\theta$: with $k = 0.5$ it is 0.08 at 30° against 1.33 for P-impedance, and over the near and mid angles it has almost the same shape as the S-impedance term, so the two trade off. Density needs far angles, beyond about 40°, and clean data. It is the least constrained of the three.

The derived products inherit these limits: $V_{\mathrm{P}}/V_{\mathrm{S}}$ depends on S-impedance as much as on P-impedance, $\mu\rho$ is S-impedance squared, and $\lambda\rho = Z_{\mathrm{P}}^2 - 2Z_{\mathrm{S}}^2$ subtracts two uncertain numbers. In Figure 7.4, with five stacks out to 35° and noise at 25% of the signal (a signal-to-noise ratio of 4), the inversion returns 96% of the P-impedance detail its band carries, 87% of the S-impedance and none of the density; the data leave all of the prior’s uncertainty in density. Out to 45° with 5% noise, density comes back, about a third of it.

Every product also has the resolution of the survey’s band. The inversion cannot return detail the band does not carry: a thin layer is smeared and loses part of its contrast, and below the band’s low cut every volume holds only the low-frequency model built from wells, which knows what the wells found and nothing else.

Reading lithology, porosity and fluid

Each rock property is read from the product most sensitive to it, usually on a crossplot of two products with the rock-physics template of Section 7.2 drawn over it:

• Lithology from $V_{\mathrm{P}}/V_{\mathrm{S}}$ or $\mu\rho$: shale has a high velocity ratio (2.0 to 2.4 in the figure’s shale), clean sand a lower one, and sand’s rigid grain frame gives it a higher $\mu\rho$.
• Porosity from P-impedance, within one lithology and one fluid: more pore space, lower $Z_{\mathrm{P}}$. In the figure’s brine sand a 5% error in P-impedance near 7.0 km/s·g/cm³ is about 1.3 porosity units.
• Fluid from $V_{\mathrm{P}}/V_{\mathrm{S}}$ or $\lambda\rho$: gas lowers the bulk modulus and with it $V_{\mathrm{P}}$, $V_{\mathrm{P}}/V_{\mathrm{S}}$ and $\lambda\rho$, while S-impedance changes little.

The simplest reading is a set of cut-offs: lines on the crossplot that divide it into classes. Figure 7.4 reads shale above $V_{\mathrm{P}}/V_{\mathrm{S}}$ = 1.94, gas below 1.68, and brine or oil sand between. Counted voxel by voxel against its earth, that reading gets the lithology wrong in 695 of 14 641 voxels (4.7%), finds 193 of the 369 gas voxels (52%), and calls 44 voxels gas that hold none, 38 of them oil: 6.2% of the section is misread. A cut-off is a decision rule, and a decision rule has an error rate. Every map read from inversion products should carry its error rate, counted at blind wells or on a model like this one.

The error rate depends on where the cut-off was set, and on what it was set on. In the figure’s gas sand the logs say $V_{\mathrm{P}}/V_{\mathrm{S}}$ = 1.59, but half the inverted gas voxels lie above 1.68, for two reasons. The band smooths a gas leg at most 30 ms thick into the rock around it: a perfect inversion in this band averages 1.675 across the gas leg and, read with the same cut-offs, finds 264 of the 369 gas voxels. And the prior, built from wells that found the sand wet, allows density almost no change, so the gas sand’s density drop is put into S-impedance instead; that raises the inverted ratio further, to 1.698 on average, and the read finds 193. The prior’s pull grows as the data improve: with stacks out to 45° and 5% noise, P-impedance and S-impedance are almost perfect and the read finds only 114. A gas cut-off placed just above the logs’ value, at 1.62, finds 65. Choose cut-offs on the inverted products at the wells, the data they will be applied to, and remember that a prior from wet wells biases an inversion against gas.

What the products cannot tell you: saturation

It is tempting to turn $V_{\mathrm{P}}/V_{\mathrm{S}}$ into a water-saturation volume. Gassmann (Section 5.3) says why that fails for gas. With the gas mixed through the pores, 5% of gas already cuts the fluid’s bulk modulus from 2.95 to 0.76 GPa, three quarters of the way to gas’s 0.05 GPa (the book’s catalogue gas, as in Figure 5.5; Figure 5.3’s gas at 2 km, used in Figure 7.2, is softer, 0.041 GPa, and takes the mixture to 0.64 GPa), so $V_{\mathrm{P}}$ and $V_{\mathrm{P}}/V_{\mathrm{S}}$ fall almost all the way at low saturation. In the figure’s sand at the crest, $V_{\mathrm{P}}/V_{\mathrm{S}}$ is 1.94 wet, 1.65 with 10% gas and 1.59 with 80%; the cut-offs find 150 gas voxels at 10% and 193 at 80%. What separates fizz gas from a commercial column is density, 2.18 against 2.00 g/cm³, and density is the product the inversion delivers worst.

So the honest fluid product of an inversion is a fluid class, gas or not, with its error rate or its probability (Section 7.5), not a saturation. Saturation comes from wells, from density where far angles and clean data allow it, or from other evidence such as a flat spot at the contact or the amplitude’s conformance to structure.

What to look for in Figure 7.4

• Choose Lithology and fluid, read. The earth (a) has a gas leg at the crest over an oil leg over brine; the read (b) finds the core of the gas leg, calls the oil brine or gas, and misreads voxels mostly along the edges of the sands, where the band smears the contrast.
• Choose Density. The gas leg is 9% lighter than the wet sand and plain in (a); the inverted density in (b) and the log in (c) hardly leave the low-frequency model. Then take the farthest angle to 45° and the noise to 5%: the gas leg appears, faintly. Watch the gas row as you do it: the read finds fewer gas voxels, not more, because the prior from the wet wells holds density near the background and the gas’s density drop goes into S-impedance.
• Cut the farthest angle to 20°. S-impedance falls to about a third of its detail, the velocity ratio copies P-impedance, the shale and sand clouds in (d) merge, and a quarter of the section is misread.
• Raise the noise to 60%. P-impedance keeps most of its detail; S-impedance loses about half, and the misread share triples.
• Lower the top of the band to 25 Hz. The band is then flat only to 20 Hz, whose quarter wavelength is 25 ms of two-way time against the thin sand’s 10 ms: the sand keeps 17% of its own impedance contrast instead of 85%, and every edge of the reservoir softens.
• Drop the gas saturation to 10%. The read changes little, 150 gas voxels found instead of 193: seismic says where gas is, not how much.
• Choose Porosity, read. The line fitted on the wells’ wet sands reads brine sand within about 0.6 porosity units and reads the oil and gas sand more than 2 units too porous, because hydrocarbons lower P-impedance just as extra porosity does.

Transform families used in industry

Cut-offs are the simplest of the rules that turn elastic products into rock properties. Three families are in use:

• Regression. Fit a line or plane in elastic space by least squares on well logs, for example $\phi = a + b\,Z_{\mathrm{P}} + c\,V_{\mathrm{P}}/V_{\mathrm{S}}$. Simple, explainable and robust to moderate noise, but it assumes one relationship everywhere; the figure’s porosity line, fitted on wet sand, is biased in the gas. Fit one per lithology and fluid.
• Template lookup. Grid the crossplot and give each cell the rock properties of the well samples or rock-physics model that fall in it. It captures non-linear relationships, but cells the wells never sampled are guesses. Hard classification with cut-offs or polygons is the simplest template.
• Machine learning. Train a network on logs to map products (and depth, attributes) to properties. It can capture any relationship the wells sample and overfits when they are few; its predictions in parts of elastic space the wells never reached are unreliable. Validate on blind wells.

Whichever is used, its output inherits the inversion’s errors and adds its own. Running two families and comparing them shows where the answer depends on the method.

Net pay: combining the readings

The asset team’s integrated product counts a voxel as net pay when it passes three cut-offs:

with cut-offs such as $V_{\mathrm{cut}} = 0.35$, $\phi_{\mathrm{cut}} = 0.10$ to $0.15$ and $S_{\mathrm{w,cut}} = 0.5$ to $0.6$, chosen per reservoir from cores and economics. Summed down each trace, net-pay voxels give a net-pay thickness map, the input to volumetrics (Section 7.6):

where $A$ is the area, $h_{\mathrm{net}}$ the net-pay thickness, $\bar{\phi}$ and $\bar{S}_{\mathrm{w}}$ averages over net pay, and $B_{\mathrm{o}}$ the formation volume factor. From seismic alone the saturation test is in practice a fluid-class test (hydrocarbon present or not), with the saturation taken from wells. A voxel counts as pay only if all three readings are right, so the errors compound: the net-pay map is less reliable than any of the maps it is made from.

How errors propagate

Each reading inherits the error of the product it is read from, and each product its own constraint, so the error rates differ. In Figure 7.4’s starting state:

• Porosity from P-impedance, the best-constrained product, is read within about 0.6 porosity units in the brine sand. Its largest error is not noise but the fluid: more than 2 units too high in the oil and gas.
• Lithology from $V_{\mathrm{P}}/V_{\mathrm{S}}$ is wrong in about 5% of voxels, mostly at the edges of sands, where the band smears the contrast, and in shale whose ratio lies near the cut-off. With near angles only it is wrong in a quarter of the section, because $V_{\mathrm{P}}/V_{\mathrm{S}}$ then rests on a poorly constrained S-impedance.
• Fluid from $V_{\mathrm{P}}/V_{\mathrm{S}}$ finds about half of the gas voxels (a perfect inversion in the same band would let it find 72%) and calls some oil gas; oil and brine cannot be told apart with these cut-offs.
• Saturation is not resolved at all by the velocity ratio, and only weakly by density at far angles.

The practical rule: deliver each rock-property map with its error rate, or better its probability (Section 7.5). A porosity map shown without its uncertainty will be trusted most where it deserves it least.

Common pitfalls and how to avoid them

• Setting cut-offs on the logs. The inverted volume has the band’s resolution and the low-frequency model’s bias; log values are often never reached. Set cut-offs on the inverted products at the wells.
• One transform for a heterogeneous reservoir. A single regression for the whole section mixes relationships that differ between shale and sand, and between fluids. Classify first, then apply a transform per class.
• Extrapolating beyond the wells. A voxel whose products fall outside the range the calibration wells sampled is an extrapolation; flag or mask it.
• Calibrating on oil wells and drilling for gas. Fluid-substitute the logs with Gassmann to the fluid you expect before calibrating, and check against any analogue.
• Trusting the density volume. Unless the stacks reach far angles with clean data, inverted density says little the low-frequency model and the prior did not, and an inversion that ties density to P-impedance will draw density anomalies wherever P-impedance changes. A density prior built from wet wells does the opposite harm: it keeps a gas sand’s density drop out of the density volume and pushes it into S-impedance, so the gas reads less gassy.
• Presenting saturation from seismic. $V_{\mathrm{P}}/V_{\mathrm{S}}$ falls almost fully at a few per cent gas, so a saturation volume read from it is a fluid-class map with made-up numbers. Deliver the class and its error rate.
• Over-trusting the hard facies map. Crisp class boundaries look certain; the read is wrong in a measurable share of voxels. Show the error rate or the probabilities with the map.