Module 02 · The rock
Module 01 left the pore space empty and ended on a result: anything that changes the bulk modulus K without changing the shear modulus G moves Vp and leaves Vs exactly where it was. A pore fluid is such a thing. It resists being squeezed and offers nothing at all against being sheared. This module puts brine, oil and gas into the frame built in module 01, works out what each does, and shows why the first few percent of gas does almost all of the work.
The same rock with four different things in its pores — empty, brine, oil, gas: what each does to Vp and to Vs
Five controls — the last one acts in steps 4 and 5
Five steps, from what a fluid is elastically to where its effect stops being large enough to see. Every number quoted in the text is read off the running page.
A fluid has a bulk modulus and no shear modulus. Squeeze water in a sealed container and it resists, weakly but measurably. Push sideways on the top of a body of water and it simply flows: there is no restoring force, so G is zero. That is the definition of a fluid rather than a property some fluids happen to have.
The three pore fluids that matter are brine, oil and gas, and their bulk moduli are not close to one another. At the pressure and temperature set here, brine is roughly 2.7 GPa, oil is about half that, and gas is around 0.05 — some fifty times softer than brine. On a linear axis the gas bar is invisible, which is why the left-hand panel is drawn that way rather than on a compressed scale.
Density separates them too, though far less dramatically: brine near 1.0 g/cm³, oil near 0.8, gas near 0.18. Both properties will matter, because a fluid changes the rock's density as well as its stiffness.
None of these is a constant. Gas in particular is not one number at all: its bulk modulus rises steeply with pressure, so gas at 5 MPa and gas at 50 MPa are different materials as far as a seismic wave is concerned. The right-hand panel follows all three with pressure.
1 Bulk modulus of the three fluids and of the mineral, for scale
2 How they change with pressure gas is the one that moves
Squeezing a saturated rock squeezes the pore space, and the fluid in the pores resists that. The rock is therefore harder to compress with a fluid in it than without, and its bulk modulus rises. How much it rises is given by Gassmann's relation, which takes the dry frame's bulk modulus, the mineral's, the fluid's and the porosity, and returns the saturated bulk modulus.
Shearing a saturated rock is a different matter. A shear deformation changes the shape of the pore space without changing its volume, so the fluid is not compressed and has nothing to push back with. Gassmann's relation says the shear modulus of the saturated rock equals the shear modulus of the dry frame, exactly. The right-hand panel is four identical bars, and that is the point of it.
A stiffer fluid raises K more, so brine raises it most and gas raises it least. Notice how much of the rise comes from a fluid that is itself far softer than the frame: brine at 2.7 GPa lifts the frame's bulk modulus by more than that, because the fluid is not acting in parallel with the frame but resisting the collapse of the pore space the frame would otherwise allow.
1 Bulk modulus, dry then saturated a stiffer fluid lifts it further
2 Shear modulus, the same four cases four identical bars
Both velocities come from the moduli and the density, as in module 01: Vp from K + 4G/3 over density, Vs from G over density. The fluid moved K and it moved the density, so both velocities change, but for different reasons and by different amounts.
Vp responds to the fluid directly. Brine gives the stiffest rock and the highest Vp; gas gives the softest and the lowest. The spread between them is the whole basis of using amplitudes to say something about fluids.
Vs responds only through the density. G is identical in all four cases, so any change in Vs is the square root of a density ratio and nothing else. Gas is the lightest fluid, so a gas-filled rock has the highest Vs of the three saturated cases. The shear wave responds to the reduced mass rather than to the gas itself.
That difference in behavior is what makes the pair useful. Vp falling while Vs holds steady or rises slightly is a signature a fluid change produces and a porosity change does not, because a porosity change moves K and G together.
1 Vp and Vs for the four cases empty, brine, oil, gas
2 Density and Vp/Vs for the same four where the shear change comes from
A pore space is rarely all gas or all brine. The gas saturation is the fraction of the pore volume the gas occupies, and the question is how the rock behaves in between.
Density behaves as expected: it falls in a straight line as heavy brine is replaced by light gas. The bulk modulus does not. When gas and brine share a pore, a squeeze compresses the gas first, because the gas is fifty times easier to compress. The mixture is therefore about as soft as the gas almost as soon as any gas is present, and the fluid modulus collapses within the first few percent of saturation.
The consequence is on the right. Vp falls steeply at low gas saturation and then almost flattens. At the settings this module opens with, taking the gas saturation from zero to five percent delivers about 87% of the entire velocity drop that going all the way to 100% gas would produce.
A seismic amplitude therefore cannot distinguish a commercial gas column from a few percent of gas dissolved out of solution. That ambiguity has a name in the industry — fizz gas — and it has been responsible for a great many dry holes drilled on convincing bright spots. Nothing later in this set removes it.
One further detail is visible if the curve is followed all the way across. Vp reaches a minimum partway along — near 32% gas at the settings this module opens with — and then rises slightly toward full gas saturation, because the fluid modulus has stopped falling while the density keeps falling.
1 Fluid modulus against gas saturation with the straight-line density for comparison
2 Vp against gas saturation and where it stops falling
The size of the fluid effect is not a constant. Two things reduce it, and both are ordinary conditions rather than edge cases.
Both panels plot the brine-minus-gas separation as a percentage of the brine velocity rather than in meters per second. In absolute terms the separation hardly changes with porosity at all — around 410 m/s in a 15% sand and 460 in a 35% one. What changes is how large that is compared with how fast the rock already is, and a reflection responds to a relative contrast rather than to an absolute velocity shift. A 400 m/s difference on a 3500 m/s rock is a much weaker signal than the same 400 on a 2500 m/s rock.
Low porosity. A tight rock has little fluid to act through and a stiff frame that dominates what fluid there is, while also being fast. The separation falls from about 18% of the brine velocity at 35% porosity to about 11% at 15%.
Depth. Gas stiffens under pressure. At 5 MPa its bulk modulus is under a hundredth of a gigapascal; by 50 MPa it is around twenty times that. A deep gas sand is elastically less different from a brine sand than a shallow one, because the gas in it is less different from brine. Here the separation falls from about 25% to about 13% across that pressure range, and this trend is the stronger of the two.
Both point the same way. The rocks in which a fluid is easiest to detect are shallow, young, porous and poorly consolidated — which is also where the frame is softest, so the fluid carries a larger share of the rock's stiffness.
1 Brine minus gas, against porosity at the pressure and temperature you set
2 Brine minus gas, against pressure at the porosity you set
Fluid substitution. A well is logged in whatever fluid it happened to encounter. Gassmann's relation run in reverse strips that fluid out to recover the dry frame, and run forwards puts a different one in. That is how a brine well is used to predict what the same sand would look like with gas, and it is the standard first step of any amplitude study.
Deciding whether a prospect can be tested with amplitudes. The panels in step 5 are, in effect, a feasibility test. If the brine and gas cases differ by only a few tens of meters per second, no amount of processing or analysis will separate them, and that can be established before the money is spent rather than after.
Interpreting a bright spot. Step 4 is the reason a bright amplitude is not a volume. The saturation curve is nearly flat above a few percent gas, so amplitude alone constrains the presence of gas far better than it constrains how much.
Knowing which rocks to be careful with. The assumption that a fluid leaves the shear modulus alone is what the whole calculation rests on. In shales, in carbonates with complex pore shapes, and where a fluid reacts chemically with the grains, it holds less well, and results built on it are correspondingly less certain.
Each answer is a number read off this page at the settings described.
Fluid properties are the Batzle-Wang equations rather than three constants, because the whole subject of this module is that fluid properties move. Brine is at 35 000 ppm salinity, oil is 32° API with no dissolved gas, and gas has a specific gravity of 0.65. Each is evaluated at the pressure and temperature on screen.
The dry frame is the module 01 soft-sand model: Hertz-Mindlin at a critical porosity of 40% with nine contacts per grain, then a modified Hashin-Shtrikman lower bound. The pressure control moves the frame and the fluids together, which is what burial does.
Gassmann's relation is used exactly as published, with the shear modulus of the saturated rock set equal to that of the dry frame. Everything in this module rests on that equality, and its assumptions are these: the pore space is fully connected, the rock is isotropic and homogeneous at the scale of the measurement, the fluid does not react with the grains, and — most importantly — the wave is slow enough that pore pressure equalizes throughout the rock during each cycle. Seismic frequencies generally satisfy that last condition; laboratory ultrasonic measurements and sonic logs often do not, which is why a log-derived velocity and a Gassmann-predicted one need not agree.
The mixture in step 4 is a Reuss average of brine and gas, which assumes the two are mixed evenly at a scale far smaller than the wavelength. That is the standard assumption and it produces the steep curve shown. Gas concentrated in patches larger than the diffusion length behaves closer to a Voigt average, which is nearly a straight line, and real rocks fall somewhere between. Brie's equation is the usual empirical way of expressing that middle ground. The choice matters: the patchy case would make a bright amplitude a much better guide to saturation than the uniform case allows. Where this page offers a patchy option it takes the Voigt average of the two fluid moduli and then runs Gassmann once, which is the simpler of the two conventions. The stricter definition averages the saturated rock moduli of the fully brine and fully gas cases, and the two do not give quite the same curve; the Rock Physics Handbook uses the second.
Deferred or absent:
Next up
Modules 01 and 02 built a rock and put a fluid in it. Module 03 follows it the rest of the way: two rocks meeting at a boundary, a reflection coefficient, a wavelet, and the trace that comes out.
Go to module 03 →