Patchy Saturation
Learning objectives
- Explain when the uniform assumption fails: patches larger than the pressure-diffusion length, or high frequency
- Contrast the uniform (Wood, lower) and patchy (upper) trends of the effective fluid modulus
- Use the Brie equation as the standard interpolation between the two limits
- Recognize that one saturation gives a range of velocities, a real 4D and DHI ambiguity
When Pressure Cannot Equalize
Wood's harmonic average earned its authority from one assumption: that the fluids are mixed finely enough for pore pressure to equalize in the time a wave takes to pass. Relax that assumption and the rule changes. Suppose the gas gathers into patches larger than the distance pressure can diffuse in a single wave period, or, equivalently, the wave frequency is high enough that pressure cannot travel between patches before the wave has gone by. Now each patch is trapped at its own pressure. A gas patch stays soft, but a neighboring brine patch is not dragged down with it, so the rock as a whole responds stiffer than the uniform mixture would. The effective fluid follows an upper trend that sits well above Wood.
The Brie Interpolation
The two limits bracket the truth: uniform mixing is the lower bound, where every phase shares one pressure, and fully patchy saturation is the upper bound, a volume-weighted arithmetic average of the phase moduli. Real rocks sit between, and the standard empirical bridge is the Brie et al. (1995) equation, , with the exponent near 3 for typical patch geometries. A smaller exponent leans toward the stiff, patchy upper trend; a larger one leans toward the soft, uniform limit. The spread is not subtle: at 20 percent gas the uniform Wood modulus is about 0.28 GPa while the Brie curve at exponent 3 reads about 1.58 GPa, more than five times stiffer, from nothing but a change in how the same gas is distributed.
One Saturation, Many Velocities, and the Part Closes
That fivefold spread is a real ambiguity, not a modeling nicety. The same gas saturation produces different velocities depending on the patch scale, so a bright amplitude or a 4D difference can record a change in how the fluid is distributed rather than how much of it is present. A patchy high-saturation gas sand and a uniform low-saturation one can imitate each other, and telling them apart is one of the standing hard problems of direct hydrocarbon indication and time-lapse monitoring.
With that, Part 3 is complete. The pore fluids have been given their properties (Batzle-Wang brine, oil, and gas at reservoir conditions), and their mixtures have been given their rules (Wood at the uniform limit, patchy above it). One thing is still missing, and it is the largest thing of all: how the fluid, carrying its bulk modulus and its density, actually couples to the solid frame to set the saturated rock's modulus and velocity. That coupling is Gassmann's theory, and it is the whole of Part 4.
References
- Mavko, G., Mukerji, T., & Dvorkin, J. (2009). The Rock Physics Handbook (2nd ed.). Cambridge University Press.
- Brie, A., Pampuri, F., Marsala, A. F., & Meazza, O. (1995). Shear sonic interpretation in gas-bearing sands. SPE Annual Technical Conference and Exhibition, SPE-30595-MS.