AVO classes and the intercept-gradient crossplot

Where a wet section plots

Take a section of shale and sandstone with brine in every pore. Neighbouring layers differ, but they differ the way such rocks always do: $V_{\mathrm{S}}$ follows $V_{\mathrm{P}}$ along the mudrock line in shale (Castagna, Batzle and Eastwood 1985) and along the sandstone line in sand (Greenberg and Castagna 1992), and density follows $V_{\mathrm{P}}$ by Gardner-type rules. A contrast across an interface is then a contrast in $V_{\mathrm{P}}$ with its usual companions, and the linearised equations of Section 5.4 tie $G$ to $R_0$. Castagna, Swan and Foster (1998) showed that such a wet background plots along a line through the origin, where two identical layers sit, with a slope set by the background $V_{\mathrm{P}}/V_{\mathrm{S}}$.

Figure 5.5 builds such a section from these rules: 360 interfaces with the scatter real rocks have about the lines, each point computed from its two layers. Its least-squares trend through the origin has a slope of $-1.61$ in the starting state, and anything from $-0.94$ to $-2.94$ as the sand and shale velocities change. There is no universal slope, so the trend is fitted to the local data, one interval at a time. The cloud also has a width: here the points scatter about the trend by $\sigma = 0.094$ in $G$, and a departure means something only when it is large against that width. That width is not one bell-shaped scatter: the cloud is four families of interface, shale on shale and sand on sand along the trend, shale on brine sand (the wet top of a sand) about $1.4\,\sigma$ below it and brine sand on shale about $1.3\,\sigma$ above. So $\sigma$ is a ruler rather than a probability: no wet interface in the starting state lies more than $2.2\,\sigma$ from the trend, and a sand’s top sits a little below it even with brine.

What a hydrocarbon does

Replace some of the brine in a sand by gas or oil and Gassmann (Section 5.3) lowers its bulk modulus, so $V_{\mathrm{P}}$ falls; the density falls a little, and $V_{\mathrm{S}}$ rises slightly, because the shear modulus is unchanged while the rock is lighter. In Figure 5.5’s starting sand, 80% gas lowers $V_{\mathrm{P}}$ by 15%, raises $V_{\mathrm{S}}$ by 5% and lowers the density by 10%.

The top of the sand, shale over sand, becomes more negative in both $R_0$ and $G$: its point moves down and to the left, below the trend. The base, sand over shale, moves up and to the right, above it. In the linearised equations swapping the layers only changes the sign of every contrast, so in those equations the base is exactly the top’s mirror image through the origin (the exact Zoeppritz amplitudes are mirror images only at normal incidence, and nearly so to 30°: in the starting state $-0.197$ for the top and $+0.214$ for the base at 30°): tops of hydrocarbon sands plot below the wet trend and bases above it (Castagna, Swan and Foster 1998). In the figure’s starting state 80% gas moves the top from $1.3\,\sigma$ to $4.5\,\sigma$ below the trend; 80% oil, a stiffer fluid, moves it to $3.0\,\sigma$.

Most of the move happens with the first few per cent of gas. With the fluids mixed in each pore, the softest fluid dominates: 10% gas cuts the pore fluid’s bulk modulus from 2.95 to 0.43 GPa, and the starting sand’s from 12.1 to 6.7 GPa, already close to its dry frame’s 5.6 GPa, so more gas has little left to change: at 10% gas the top already sits $3.8\,\sigma$ below the trend. This is the fizz-water problem (Domenico 1976): the crossplot shows that a sand holds gas, not how much.

The AVO classes

The classes name where the top of a hydrocarbon sand falls, which depends above all on how the sand’s impedance compares with the shale above it (Rutherford and Williams 1989). In SEG normal polarity a positive $R_0$ is a peak and a negative one a trough. With the thresholds this book uses here and in Section 5.4:

• Class I, $R_0 \ge 0.04$ and $G < 0$: a peak that dims with angle, and with a steep gradient changes sign before 30°. The sand is harder than its shale, typically compacted or cemented.
• Class II, $|R_0| < 0.04$ and $G < 0$: faint at normal incidence, its amplitude at far angles set by its negative gradient. The sand and the shale have about the same impedance. Ross and Kinman (1995) split off Class IIp, a small positive intercept, $0 < R_0 < 0.04$, that changes sign with angle (here, before 30°, which is $G < -4R_0$); on a full stack its near and far halves cancel and the sand can vanish.
• Class III, $R_0 \le -0.04$ and $G < 0$: a trough that brightens with angle. The sand is softer than its shale: the classic bright spot of young, poorly consolidated sands.
• Class IV, $R_0 < 0$ and $G > 0$: a trough that dims with angle. A soft sand lies under a seal that is faster in shear, such as a hard shale or a tight siltstone (Castagna and Swan 1997). Its amplitude falls with offset, so a screen for amplitude that grows with offset misses it, although the point still lies below the wet trend.

The thresholds are conventions, and authors differ. The quadrant $R_0 > 0$, $G > 0$ has no class: it is where the bases of Class III and IV sands plot. A class describes a reflection, not a fluid: a brine sand has one too (the starting sand of Figure 5.5 is Class II with brine and Class III with gas), and the class follows the sand’s stiffness against its seal, which changes with burial. What flags the hydrocarbon is the departure from the wet trend.

What to look for in Figure 5.5

• Raise the saturation from 0. The top and base leave the cloud on mirror-image paths that bend early: most of the move is done by 10%.
• Stiffen the sand to 4200 m/s. Gas now lowers $V_{\mathrm{P}}$ by 3%, not 15%, because a stiff frame barely feels the pore fluid; the top is a Class I peak that dims with angle, and it departs less from the trend.
• Set the sand to 3600 m/s. The top is a small peak that changes sign at 17°: Class IIp, which the gathers show and a stack would hide.
• Put a soft sand (2500 m/s) under a hard shale (3500 m/s). The top is a Class IV trough that dims with angle, yet it sits $5.0\,\sigma$ below the trend.
• Add the rock library (all but its oil and gas sands, which the figure’s own sand stands for). With no hydrocarbon anywhere, coal plots almost as far below the trend as the gas sand, and halite and anhydrite plot off it on either side: a departure can be lithology.

How interpreters use the crossplot in practice

A typical workflow on real data:

• Compute $R_0$ and $G$ volumes from amplitude-preserved prestack gathers, by fitting amplitude against $\sin^2\theta$ at every sample (or by prestack inversion).
• Crossplot the two volumes over one stratigraphic interval. The result is a dense cloud along the wet trend with a tail of outliers.
• Fit the wet trend to that interval’s data, not to a textbook value, and measure the cloud’s width about it.
• Define an anomaly polygon below the trend for tops (and its mirror above for bases), from a rock-physics model and well control.
• Paint every sample inside the polygon back onto the seismic. Tops and bases of a real hydrocarbon sand should appear as pairs, and should conform to a trap.
• Tie to wells, and look for independent evidence, a flat spot or a change at the contact, before calling the fluid.

Beyond the intercept and gradient: lambda-rho and mu-rho

The intercept and gradient are one way to parameterise the same elastic contrasts. Goodway, Chen and Downton (1997) used the P and S impedances from prestack inversion to form $\lambda\rho = Z_{\mathrm{P}}^2 - 2Z_{\mathrm{S}}^2$ and $\mu\rho = Z_{\mathrm{S}}^2$. The second responds mainly to the rock frame, so to lithology; the first falls sharply with gas, because gas lowers the bulk stiffness and hence $\lambda$. Crossplotted, the two separate fluid from lithology more cleanly than $R_0$ and $G$, at the cost of a full inversion. The physics is the same; only the axes change.

Common crossplot pitfalls

• Assuming the trend. Its slope depends on the local rocks; in Figure 5.5 alone it runs from $-0.94$ to $-2.94$. Fit it to each interval’s data.
• Mixing lithologies. Coal, carbonates and evaporites do not follow the clastic wet trend. A crossplot over several lithologies has several backgrounds, and a lithology contrast can sit as far off the trend as a gas sand.
• Reading noise as a trend. Uncorrelated noise in the gathers makes the errors of the fitted intercept and gradient anticorrelated, so noise smears each point along a line of negative slope (Cambois 2000). A noisy cloud can look like a background trend, or hide one.
• Reading saturation from distance. A little gas departs most of the way that a lot does; the crossplot cannot tell fizz water from a commercial column.
• Trusting amplitudes the processing did not preserve. $R_0$ and $G$ are only as good as the gathers: the angle each trace really has, the NMO stretch at far offsets, tuning in thin beds and the processing’s amplitude balance all move points.
• Forgetting how much volume a point stands for. A dense part of the cloud is many samples; an isolated point may be one sample of noise. Check how much of the volume an anomaly occupies, and where.