The physics of fluid diffusion in anisotropic media was studied, based on Biot's theory of poroelasticity and using wave propagation concepts. Diffusion and elastic strain can be uncoupled fully, being a good approximation in many situations. We have used a correction to the stiffness of the rock under conditions of transverse isotropy and uniaxial strain to model borehole conditions. The concepts of phase, group, and energy velocities were analyzed to describe the location of the diffusion front, and the attenuation and quality factors were obtained to quantify the amplitude decay. We have found that the location of the front is described correctly by the energy velocity. The Green's function in anisotropic media can be obtained by applying a change of coordinates to the isotropic solution. We have simulated the diffusion in inhomogeneous media using a time-domain spectral explicit scheme and the staggered Fourier pseudospectral method to compute the spatial derivatives. The method is based on a spectral Chebychev expansion of the evolution operator of the system. The scheme allows the solution of linear periodic parabolic equations, having accuracy within the machine precision, in time and in space. The results match the analytic solution obtained from the Green's function. The performance of the algorithm is confirmed in the case of a pressure field generated by a fluid-injection source in a hydrocarbon reservoir where the properties vary fractally.