Mixing Fluids

Part 3, Part 3: The Fluids

Learning objectives

State the uniform fine-scale mixing assumption: every fluid phase feels the same pressure
Compute the effective fluid modulus as the Wood harmonic average of the phase moduli
Show that the harmonic average is ruled by the softest phase, so a little gas crashes the mixture
Contrast the collapse of the effective modulus with the near-linear volume average of density

Two Fluids, One Pore

The pore space rarely holds a single fluid. A gas cap grades into an oil column, an oil leg sits above brine, a producing reservoir carries gas coming out of solution: two fluids share the pores, and the rock needs one effective fluid to stand in for the pair. The answer depends on a physical question that Part 2 already framed: at what scale are the fluids mixed? Take the case where they are intermixed finely, at the scale of individual pores, so that when a wave squeezes the rock the pore pressure equalizes across the mixture and every phase feels the same stress. That isostress condition is exactly Wood's suspension picture, and it fixes the effective modulus.

Wood's Harmonic Average

Under uniform mixing the effective fluid bulk modulus is the harmonic, or Reuss, average of the phase moduli, weighted by saturation: $\dfrac{1}{K_{fl}} = \dfrac{S_1}{K_1} + \dfrac{S_2}{K_2}$ fl=dfracS_1K_1+dfracS_2K_2. A harmonic average is not a compromise between its terms; it is dragged down toward the smallest of them, because the softest phase does most of the yielding under a shared stress. With brine at 3.03 GPa and gas at 0.061, mixing in just 5 percent gas drops the effective modulus from 3.03 to about 0.88 GPa, and 10 percent gas drops it to about 0.51 GPa, a sixfold collapse from the brine value. The mixture is already behaving far more like gas than like the brine that still fills nine tenths of the pore.

Density Barely Notices

Density tells the opposite story. It is the simple volume average, $\rho_{fl} = S_1 \rho_1 + S_2 \rho_2$ fl=S_1rho_1+S_2rho_2, a straight line in saturation with no collapse in it. That same 10 percent gas moves the mixture density only from 1.068 g/cc to about 0.98, a few percent. So the two properties the fluid controls behave asymmetrically: the bulk modulus falls off a cliff while the density strolls down a ramp. Because the P-wave carries the modulus in its numerator and the density in its denominator, the modulus collapse wins, and a small gas saturation produces a large drop in Vp. This asymmetry is why partial gas saturation is so seismically loud, and it is also a warning: a bright spot can mean a pore full of gas or a pore barely touched by it.

All of this rested on one assumption, that the fluids are mixed finely enough for pressure to equalize. If instead the gas gathers into patches too large to bleed their pressure into the surrounding brine, the rock responds stiffer and the harmonic average is the wrong rule. That patchy case is the closing section of the part.

References

Mavko, G., Mukerji, T., & Dvorkin, J. (2009). The Rock Physics Handbook (2nd ed.). Cambridge University Press.
Wood, A. B. (1955). A Textbook of Sound (3rd ed.). G. Bell and Sons.