Material anisotropy may significantly influence the behavior of fluid transport in sedimentary basins and other environments with laminated or foliated rocks. In this paper, we present a fracture mechanics model to investigate primary migration of petroleum (oil and gas) through propagation of a vertical, buoyancy-driven blade crack in a transversely isotropic source rock. The source rock is assumed to have very low permeability and hence can be modeled as a linear elastic medium. Fracture parameters (stress intensity factor and crack opening displacement) are derived using an equivalent set of anisotropic elastic properties. The crack propagation velocity (i.e., petroleum migration velocity) and crack opening (fracture aperture) are determined using a fracture mechanics criterion together with the first order approximation of plane Poiseuille flow equations of fluid mechanics. For subhorizontal layering, we find that the fluid migration velocity and fracture aperture are significantly increased if the elastic modulus in the vertical direction is smaller than that in the horizontal direction. Finally, we discuss the applicability of the formula for isotropic materials along with the equivalent anisotropic elastic parameters introduced in this paper to evaluate fracture aperture for cracks in anisotropic rocks.