The Xu-White Model
Learning objectives
- Read Xu-White as the workhorse inclusion frame for shaly sands: split the pore space by clay content into stiff sand pores and compliant clay pores
- Follow the recipe: build the dry frame with DEM on two pore aspect ratios, sand near 0.12 and clay near 0.03, over the Hill mineral mix, then saturate with Gassmann
- Anchor the numbers: at 25 percent porosity brine-saturated, Vp falls from about 3.40 km/s at 5 percent clay to 2.75 at 20 percent to 2.33 at 40 percent
- Explain why clay makes a rock slow twice: it is both a softer mineral and a source of compliant, crack-like pores
Two Pore Shapes, Not One
Real clastic reservoirs are not clean sand; they are mixtures of sand and clay, and the two carry porosity of different character. Xu and White made that the center of their model. They split the total pore space into two populations by clay content: the fraction associated with the sand grains is given a stiff, rounder aspect ratio near 0.12, and the fraction associated with clay is given a compliant, crack-like aspect ratio near 0.03. This is the inclusion machinery of Part 6 put to practical work. Each population is a family of oblate pores with its own shape, so the same DEM construction that built a single-pore frame now builds a two-pore frame, embedding sand pores and clay pores together into the mineral.
The Recipe
The workflow chains the tools of the last two parts. First mix the minerals: quartz for the sand fraction and clay for the rest, combined with the Hill average to get the solid modulus of the grain mix. Second build the dry frame: run DEM on that mineral, adding the empty sand pores at aspect ratio 0.12 and the empty clay pores at 0.03 up to the total porosity. Third saturate: pour brine into the finished dry frame with Gassmann, on the low-frequency route, to get the saturated moduli and then the velocities. The result at 25 percent porosity, brine-saturated, reads Vp 3.40 km/s and Vs 2.01 for a clean sand of 5 percent clay, Vp 2.75 for a shalier 20 percent clay, and Vp 2.33 with Vs 0.90 for a muddy 40 percent clay. More clay, a slower rock, at fixed porosity and fixed fluid.
Why Clay Slows a Rock Twice
The velocity drop with clay is steeper than mineralogy alone would predict, and the two-pore split explains why. Clay hurts the rock through two separate channels. First it is a softer mineral: clay's bulk and shear moduli sit well below quartz's, so replacing sand grains with clay lowers the solid stiffness the frame is built on. Second, and less obvious, clay brings crack-like pores: its share of the porosity carries the compliant aspect ratio of 0.03, and as the previous sections showed, flat pores soften a frame far out of proportion to their volume. So a shaly sand is slow twice over, once for its weaker mineral and once for its crackier pore space, and the model captures both at the same time. This is why shaly sands plot below clean-sand trends on velocity-porosity crossplots, a shift that mineral fraction by itself cannot account for. Xu-White is the everyday tool for that correction in clastic reservoirs. The final section takes the same two-aspect-ratio idea to carbonates, where the pore shapes are not sand and clay but interparticle, moldic, and crack, and shows that the carbonate velocity scatter is pore type made visible.
References
- Xu, S., & White, R. E. (1995). A new velocity model for clay-sand mixtures. Geophysical Prospecting, 43(1), 91-118.
- Mavko, G., Mukerji, T., & Dvorkin, J. (2009). The Rock Physics Handbook (2nd ed.). Cambridge University Press.