Predicting seismic velocities in isotropic fluid-saturated rocks is commonly done using the isotropic Gassmann theory. For anisotropic media, the solution is expressed in terms of stiffness or compliance, which does not provide an intuitive understanding on how the fluid affects wave propagation in anisotropic media. Assuming weak anisotropy, we expressed the anisotropy parameters of transversely isotropic saturated media as a function of the anisotropy parameters in the dry medium, the bulk and shear moduli of the saturated and dry media, the grain and fluid bulk moduli, and the porosity. By deriving an approximation of the anellipticity parameter , we discovered that if the dry medium was elliptical, the saturated medium was also elliptical but only if the porosity exceeded a certain threshold value. This result can provide a way of differentiating between stress- and fracture-induced anisotropy.