Rocks, pores and stiffness

1Grains, and the space between them

Volume fractions, and the density that follows from them

A clastic rock is a pack of mineral grains with space between them. The fraction of the total volume that is space is the porosity. The rest is solid, and in a sandstone that solid is mostly quartz with some clay.

Volume fractions are the simplest description of a rock, and one seismic property follows from them directly. Density is a volume-weighted average: each component contributes its own density in proportion to how much of the rock it occupies. With the pore space empty, the bulk density of the rock is the mineral density multiplied by the solid fraction.

Quartz and clay have different densities — 2.65 and 2.58 g/cm³ — so the clay fraction moves the mineral density, but only slightly. Porosity moves the bulk density far more, because it replaces solid with nothing.

1 What occupies the volume quartz, clay and empty space

2 Bulk density against porosity a straight line, because it is an average

Porosity—

Solid fraction—

Density of the mineral mix—

Dry bulk density of the rock—

Same rock at 5% porosity—

What is in the pore space here—

Density is the one rock property that mixes this simply. Neither of the two stiffnesses does, which is the subject of step 3.

2Two separate ways to deform a rock

Squeezing it, and sliding it

A rock can be deformed in two independent ways, and it resists them separately.

Squeezing. Press on a block of rock equally from every side and it gets smaller without changing shape. The property that resists this is the bulk modulus, written K: the pressure applied divided by the fractional change in volume it produces. A large K means a large pressure is needed for a small change in volume.

Shearing. Push the top of the block sideways while holding the bottom still and it changes shape without changing volume. The property that resists this is the shear modulus, written G. A large G means the block barely tilts.

Both are stresses divided by strains, so both have the units of a pressure. Rock moduli are quoted in gigapascals. The two panels below deform a block of the current rock under the same applied stress, so the more it moves, the softer that rock is in that mode. Nothing forces the two to move together: a rock can be much harder to squeeze than to shear, or the reverse.

1 Squeezed from every side volume changes, shape does not

2 Sheared sideways shape changes, volume does not

Bulk modulus of this rock, K—

Shear modulus of this rock, G—

Volume change under the test stress—

Shear strain under the same stress—

G ÷ K for this rock—

What each one resists—

A fluid has a bulk modulus and no shear modulus at all. That is the whole of what makes a pore fluid visible to seismic data, and it is module 02.

3The pack is far softer than the grains

Where the rock's stiffness actually comes from

Quartz is stiff. A single quartz grain has a bulk modulus of 37 GPa and a shear modulus of 44. A sandstone made of quartz grains has moduli several times smaller, and the reason is that the rock is not a solid block of quartz. It is grains touching each other at small contacts, and the load passes through those contacts.

Deforming the rock does not mean deforming the grains. It means moving the contacts: flattening them, sliding them past one another, rearranging the pack. That is much easier than compressing quartz, so the stiffness of the arrangement — the dry frame — is what the rock's moduli actually measure.

Two consequences follow, and both are visible in the panels. Porosity matters enormously, because a looser pack has fewer and smaller contacts. And the frame is pressure-dependent: pressing the grains together harder flattens the contacts and stiffens the rock, which is why the same sand is faster when it is buried deeper.

The dashed lines mark the mineral values, which the frame would only reach at zero porosity. The gap between the two curves and those lines is the difference between a rock and the material it is made of.

1 Both moduli against porosity with the mineral values marked

2 Both moduli against pressure at the porosity you set

Bulk modulus of the mineral—

Bulk modulus of the frame—

Frame as a fraction of the mineral, K—

Shear modulus, mineral → frame—

Frame K at 5 MPa → at 50 MPa—

Why the two differ so much—

The pressure here is the pressure carried by the grain contacts, which is the weight of the overburden less whatever the pore fluid is holding up. Separating those two is a subject of its own and is not taken up in this set.

4From two moduli to two velocities

And which modulus goes into which

A seismic wave is a deformation traveling through rock, so how fast it travels depends on how hard the rock resists that deformation and on how much mass has to be moved. Stiffer is faster, denser is slower, and each velocity takes a square root of stiffness over density.

A P-wave moves the rock along the direction it travels. That compresses the rock, and it also distorts it, because the material either side of the moving front is not free to spread out. So a P-wave feels both moduli, in the combination K + 4G/3, and its velocity is the square root of that over the density.

An S-wave moves the rock across the direction it travels. It changes shape and not volume, so it feels the shear modulus alone, and its velocity is the square root of G over the density.

That asymmetry is used throughout the rest of this set. Vp responds to both moduli; Vs responds to only one. Anything that changes K without changing G will move Vp and leave Vs where it was.

1 Both velocities against porosity the same rock across the porosity range

2 What goes into each velocity the stiffness term above the line in each case

K and G of this rock—

K + 4G/3, the P-wave stiffness—

Dry bulk density—

Vp and Vs that give—

Vp/Vs for this dry rock—

Which modulus each velocity feels—

A dry rock has a Vp/Vs near 1.5 everywhere these controls reach — between about 1.43 and 1.63 — which is well below any real water-bearing sandstone. Putting brine in the pores is what raises it, and that is module 02.

5Change one modulus at a time

The result module 02 is built on

The two multipliers under step 5 only scale the frame's moduli directly, holding the density fixed. They are not a physical process; they are a way of asking what each velocity is sensitive to, by moving one input and watching both outputs.

The left-hand panel scales the bulk modulus. Vp rises and Vs does not move at all — its line is exactly flat, because G is unchanged and G is the only stiffness a shear wave feels. The right-hand panel scales the shear modulus, and both velocities respond, because G appears in the P-wave stiffness as well.

The flat line has a specific meaning. Any process that acts on a rock's resistance to compression without touching its resistance to shear will show up in Vp and be invisible in Vs. A pore fluid is exactly such a process: a fluid resists being compressed and offers nothing at all against shearing.

That is the whole basis of using seismic amplitudes to say something about pore fluids, and module 02 builds it properly.

1 Scaling the bulk modulus shear velocity does not respond

2 Scaling the shear modulus both velocities respond

Vp, unscaled → scaled—

Vs, unscaled → scaled—

Change in Vs from the K multiplier alone—

Change in Vp from the K multiplier alone—

Vp/Vs, unscaled → scaled—

What this rules out and what it allows—

Density is held fixed while the moduli move, which no real process does. A pore fluid changes the density as well as the bulk modulus, and module 02 puts both back.

WWhy it matters

Where these ideas are used

Reading a velocity log. A sonic log records how fast the rock is, and the two largest controls on that number are porosity and how well the grains are cemented. A slow interval is usually a porous or poorly consolidated one rather than a different mineral.

Depth trends. The same sand is faster deeper, because the grain contacts carry more load and the frame is stiffer. That is a property of the frame rather than of the mineral, and it is why a velocity measured in one well does not transfer directly to a target at another depth.

Overpressure. Where pore fluid carries part of the overburden, the grain contacts carry less, the frame stays soft, and the rock is slower than its depth would suggest. That reversal is one of the few things seismic velocities detect directly, and it is a drilling safety issue.

Why a shear measurement is worth having. Vs responds to the frame and the density and to nothing else. Anything that changes the pore fluid moves Vp and leaves Vs almost alone, so comparing the two separates a fluid change from a rock change. Getting at Vs from P-wave seismic data is what the rest of this set is about.

Exercises

Each answer is a number read off this page at the settings described.

Compare how density and stiffness respond to porosity. With no clay and 20 MPa on the contacts, take porosity from 5% to 38% and note the dry bulk density and the frame's bulk modulus at each end.
Answer
Density falls from about 2.52 to 1.64 g/cm³, a drop of roughly a third. The bulk modulus falls from about 19.4 to 2.2 GPa, a drop of nearly ninety percent. Both respond to the same porosity, and they respond on completely different scales, because density is a volume average of what is present while stiffness depends on how the solid is connected. Adding pore space removes a little mass and destroys a great deal of structure.
Find out how much of the mineral's stiffness survives. At 25% porosity with no clay, compare the frame's bulk modulus with quartz's own.
Answer
About 4.7 GPa against 37, so the rock keeps roughly an eighth of the stiffness of the material it is made of. The missing stiffness is not lost in the grains, which are unchanged; it is in the contacts, which are small and easy to move. A rock is a structure, and this is the number that says so.
Bury the rock deeper. At 25% porosity, take the pressure on the contacts from 5 MPa to 50 MPa and watch both moduli and both velocities.
Answer
The bulk modulus roughly doubles, from about 3.1 to 6.1 GPa, and Vp rises from about 1994 to 2817 m/s. Nothing about the rock's composition changed — same grains, same porosity, same density. Pressing the contacts together harder makes them stiffer. This is why a velocity measured on one interval does not carry to a target at a different depth, and it is the mechanism behind the velocity reversal in an overpressured zone.
Establish that Vs cannot see the bulk modulus. Leave the shear multiplier at 1.00 and take the bulk multiplier from 0.50 to 2.00, watching both readouts in step 5.
Answer
Vp rises by about 31% across that range and Vs does not change by any amount at all — the readout stays at 0.0%, and the line in the left-hand panel is exactly horizontal. This is not an approximation or a small effect being neglected. Vs is the square root of G over density, and K does not appear in it. Any process acting on K alone is invisible to a shear wave.
Check the ratio for a dry rock. Read Vp/Vs at several porosities with no clay, then add clay and read it again.
Answer
It stays near 1.5 across the whole set of controls — 1.50 at 10% porosity, 1.48 at 25%, 1.47 at 30%. Clay pushes it up a little, to 1.51 at 25% porosity and 60% clay, and further at low porosity, because clay is much weaker in shear than quartz relative to its bulk modulus. The extremes the sliders can reach are about 1.43 and 1.63. Real water-bearing sandstones run nearer 1.8 to 2.0 and shales higher still, so every one of these numbers is too low. The whole of the difference is the pore fluid, which this module leaves out, and closing that gap is what module 02 does.

Key points

A rock is grains plus space. Porosity is the fraction that is space, and density is a straight volume-weighted average of what occupies the volume.
There are two independent stiffnesses. The bulk modulus K resists a change in volume; the shear modulus G resists a change in shape. Both are stresses over strains and both are quoted in gigapascals.
The rock is far softer than its grains. Load passes through small grain contacts, so the moduli of the dry frame are several times smaller than the mineral's — about an eighth at 25% porosity — and porosity controls how much smaller.
The frame stiffens under pressure. Pressing the contacts together harder makes the rock faster, with no change in composition, which is why the same sand is faster when buried deeper.
Vp feels both moduli in the combination K + 4G/3, divided by density, under a square root.
Vs feels only G, divided by density, under a square root. K does not appear in it at all.
So anything that changes K alone moves Vp and leaves Vs exactly where it was. A pore fluid is such a thing, which is why a fluid can be seen at all — and module 02 takes it up.
A dry rock has Vp/Vs near 1.5 across every setting here, well below the 1.8 to 2.0 of a real water-bearing sandstone. The difference is the fluid this module left out.

MMethod, simplifications and references

What is computed, and what is deferred

The mineral mix is quartz and clay combined by a Voigt-Reuss-Hill average, with quartz at K = 37 GPa, G = 44 GPa, ρ = 2.65 g/cm³ and clay at K = 21, G = 7, ρ = 2.58. Clay is not a single mineral and its published moduli vary widely; these are middle-of-the-road values and a different choice would move the numbers here without changing anything about their behavior.

The dry frame uses the soft-sand, or friable-sand, model: Hertz-Mindlin contact theory at a critical porosity of 40% with nine contacts per grain, then a modified Hashin-Shtrikman lower bound down to zero porosity. It is pressure-dependent and it reproduces the low velocities of young clastic sections, which is where amplitude work is usually done. It is a lower bound, so it describes an uncemented pack; a well-cemented sandstone is stiffer than this model at the same porosity, and a contact-cement or constant-cement model would be the right choice there.

The blocks in step 2 are drawn at true strain for an applied stress of 0.5 GPa. That stress is far larger than anything a rock experiences in the ground, and it is used so that the deformation is large enough to see. The comparison between the two panels, and between one rock and another, is faithful; the absolute size of the movement is not.

Deferred rather than simplified:

The pore space is empty. Every number on this page is a dry-rock number. Nothing here describes a rock below the water table, and module 02 is where the fluid goes in.
The multipliers in step 5 are not a process. They move a modulus with the density held fixed, which is a sensitivity test rather than a physical change. A real pore fluid moves K and density together.
Pressure is the pressure on the grain contacts, sometimes called the effective pressure. How it relates to burial depth and to the pore pressure is a separate subject and is not covered in this set.
Anisotropy, cracks, cementation and grain-scale texture are all absent. The pack is treated as an isotropic assembly of identical spheres, which is a long way from a real sandstone.

References

Mavko, G., Mukerji, T., and Dvorkin, J., 2020, The Rock Physics Handbook, 3rd ed.: Cambridge University Press. — the standard reference for everything on this page.
Dvorkin, J., and Nur, A., 1996, Elasticity of high-porosity sandstones: theory for two North Sea data sets: Geophysics, 61, 1363–1370. — the soft-sand model.
Mindlin, R. D., 1949, Compliance of elastic bodies in contact: Journal of Applied Mechanics, 16, 259–268.
Nur, A., Mavko, G., Dvorkin, J., and Galmudi, D., 1998, Critical porosity: a key to relating physical properties to porosity in rocks: The Leading Edge, 17, 357–362.
Hill, R., 1952, The elastic behavior of a crystalline aggregate: Proceedings of the Physical Society A, 65, 349–354. — the averaging used for the mineral mix.

The rock

Step 5 only

1Grains, and the space between them

2Two separate ways to deform a rock

3The pack is far softer than the grains

4From two moduli to two velocities

5Change one modulus at a time

WWhy it matters

Exercises

Key points

MMethod, simplifications and references

02 · Fluid in the pores