In a recent publication <Zhan and Yortsos, 2000> we proposed a direct method for the inversion of the permeability field of an isotropic porous medium based on the analysis of the displacement of a passive tracer. By monitoring the displacement front at successive time intervals (e.g., using a tomographic method), the permeability can be directly obtained from the solution of a nonlinear boundary value problem. In this paper we extend this approach to the case when the porous medium is anisotropic. When the principal axes of anisotropy are known and fixed, a procedure is proposed in which the tracer is injected two (or three) consecutive times along the two (or three) principal directions (for the case of a two-dimensional (2-D) (or three-dimensional) problem). It is shown that the diagonal components can be obtained from the solution of two (or three) coupled boundary value problems involving the experimentally obtained fields of arrival times. Numerical examples show that the method works well when the permeability variation is not very sharp (e.g., for correlated distributions). When the permeability tensor is full and the principal axes vary in space, a procedure involving the injection in three different directions (for the case of a 2-D problem) is also described. ¿ 2001 American Geophysical Union