Drainage and Imbibition

The Rock's Signature Curve

Measure capillary pressure against water saturation and the rock draws its own signature. Forcing oil in, drainage, traces it from right to left: nothing happens until the pressure beats the entry pressure $P_e$ and oil pushes into the largest pores; then a sorting plateau as the bulk of the pore space drains at nearly constant pressure; then a steep climb as only the finest pores are left, ending at the irreducible water $S_{wi}$ that no pressure can displace.

Brooks-Corey and Sorting

The Brooks-Corey law captures the shape:

The exponent $\lambda$, the pore-size distribution index, sets how flat the plateau is. A well-sorted rock, with one dominant throat size, has a high $\lambda$ and drains almost all at once on a flat plateau; a poorly sorted rock has a low $\lambda$ and bleeds down a long ramp. So the curve reads the pore-size distribution, not just the average size.

Hysteresis and the Initial State

Push water back, imbibition, and the rock does not retrace its steps. The imbibition curve runs below the drainage curve and stops short of full water, leaving a residual oil saturation trapped in pore bodies it cannot escape. That gap is capillary hysteresis. It matters which curve you use: a reservoir was filled by oil migrating up and displacing water, a drainage process, so the drainage curve, not imbibition, sets the initial water saturation the well will log. Imbibition governs what happens later, during a waterflood.

References

• Brooks, R. H. and Corey, A. T. (1964). Hydraulic properties of porous media. Hydrology Papers, Colorado State University.
• Tiab, D. and Donaldson, E. C. (2015). Petrophysics, 4th ed. Gulf Professional Publishing.