The Gassmann Idea

Part 4, Part 4: Gassmann and Fluid Substitution

Learning objectives

State Gassmann's central result: from the dry-frame moduli, the mineral, the porosity, and the pore fluid, predict the saturated moduli
Show that Gassmann adds a fluid stiffening term to the dry-frame bulk modulus and leaves the shear modulus exactly unchanged
List the four assumptions Gassmann rests on and read each as a condition for careful use
Watch the bulk-modulus bar grow by the Gassmann term while the shear bar stays fixed as the fluid changes

The One Equation of Fluid Substitution

Part 3 ended one step short. The pore fluids were given their properties and their mixing rules, but the coupling was still missing: how a fluid, carrying its bulk modulus $K_{fl}$ fl and its density, actually stiffens the rock it sits in. Gassmann (1951) supplies that coupling, and it is the most-used result in the whole subject. The claim is precise and, at first hearing, almost too generous: if you know the moduli of the dry rock frame, the mineral the grains are made of, the porosity, and the pore fluid, you can predict the moduli of the saturated rock. Four things you can measure or model going in, the saturated answer coming out.

The bulk part is one equation, $K_{sat} = K_{dry} + \dfrac{\left(1 - K_{dry}/K_{min}\right)^2}{\phi/K_{fl} + (1-\phi)/K_{min} - K_{dry}/K_{min}^2}$ dfracleft(1−Kdry/Kminright)2phi/Kfl+(1−phi)/Kmin−Kdry/Kmin2, and the shear part is shorter still, $\mu_{sat} = \mu_{dry}$ . Read the first in words before trusting a single symbol in it: the saturated bulk modulus is the dry-frame bulk modulus plus a stiffening term that the fluid contributes. Read the second the same way: the saturated shear modulus is simply the dry one, unchanged.

Two Statements, One Physical Picture

Gassmann is really two statements. The first is that the fluid stiffens the bulk modulus and never softens it. The added term is a square over a denominator that stays positive for any real frame, so it is never negative, vanishing only when there is no fluid at all. A saturated rock is therefore always at least as hard to squeeze as the same rock dry, and how much harder is set by the fluid: a stiff brine adds a great deal, a soft gas adds almost nothing. The second statement is that the shear modulus does not move, because the pore fluid has zero shear modulus and can add nothing to the rock's resistance to a change of shape (Part 1.2). The fluid reaches $K$ and it reaches $\rho$ , but it never reaches $\mu$ .

Put numbers on it with the sand this part will carry throughout: a soft clean sand at porosity $\phi = 0.28$ , with a dry frame of $K_{dry} = 6$ dry=6 GPa and $\mu = 7$ GPa, its grains quartz at $K_{min} = 36.6$ min=36.6 GPa. Dry, its bulk modulus is 6. Saturate it with brine and the Gassmann term adds 6.35, lifting $K_{sat}$ sat to 12.35, very nearly double. Saturate the same frame with gas and the term adds only 0.12, leaving $K_{sat}$ sat at 6.12, barely above the dry value. Through both substitutions the shear modulus sits unmoved at 7.

The Assumptions Are Part of the Result

The generosity has a price, and attending to it is exactly what separates careful use from careless. Gassmann holds under four assumptions. First, the rock is isotropic and the mineral is homogeneous, ideally a single mineral. Second, the pore space is fully connected and the fluid is free to move through it. Third, and most important, the process is low frequency: the wave squeezes the rock slowly enough that pore pressure equilibrates through the entire connected pore space within a single wave period, the relaxed limit introduced in Part 0.4 and met again in the patchy saturation of Part 3.6. Fourth, the fluid and the frame do not interact chemically: the fluid changes how hard the rock is to squeeze, but it does not weaken or soften the grains themselves.

Hold onto those four, because each of them returns in Part 4.5 as a way the theory can fail. For now the equation is the prize, and it comes with a practical catch. It wants the dry frame as an input, and you almost never have a dry frame in the subsurface: what a well log hands you is the rock already saturated with whatever fluid is in it. The next section turns Gassmann into the three-step workflow that gets around exactly this, stripping the in-situ fluid out to recover the dry frame before putting a new fluid in.

References

Gassmann, F. (1951). Uber die Elastizitat poroser Medien. Vierteljahrsschrift der Naturforschenden Gesellschaft in Zurich, 96, 1-23.
Mavko, G., Mukerji, T., & Dvorkin, J. (2009). The Rock Physics Handbook (2nd ed.). Cambridge University Press.