From the equations governing the deformation of a porous medium containing three fluid phases, I derive expressions for the phase velocity of the various modes of displacement. These expressions are valid for a medium with smoothly varying heterogeneity. There is a single mode of transverse displacement, similar in nature to an elastic shear wave. The four-phase velocities of the longitudinal modes of displacement are derived from the solutions of a quartic equation. The coefficients of the polynomial equation are written as linear sums of the determinants of 4 × 4 matrices. The matrices contain various combinations of the parameters from the governing equations. The three-phase expressions are compared to two-phase estimates for the case in which one of the fluid saturations vanishes. Also, in a numerical illustration, velocity variations of around 10% are associated with the cyclic injection of carbon dioxide and water into an oil-saturated reservoir.

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