The theory of finite Markov processes is used to calculate a normalized tortuosity for a porous solid with a well-defined internal structure characterized by an interconnected network of serial and parallel channels. The model introduced is based on the Menger sponge, a symmetric fractal set of dimension D=ln 20/ln 3. Numerically exact values of the mean walklength [n] for a permeant diffusing through this system are calculated both in the presence and absence of a uniform gradient (bias or external field) acting on the permeant. The ratio of walklengths is then used to define unambiguously a normalized tortuosity for the medium. Different assumptions on the initial spatial distribution of the permeant are investigated and the study is designed so that the effects of one or multiple exit sites are quantified. As expected, the tortuosity factor is dependent on system size, and quantitative results are presented for the first- and second-generation Menger sponge. Our calculations document that for a given system size: (1) the mean passage time can change by an order of magnitude depending on the number of exit pores available to the permeant; and (2) restricting the number of exit pores suppresses differences between values of [n] calculated for different initial conditions. Finally, we study whether the geometrical self-similarity of Menger sponge translates into a corresponding scaling of diffusion times for different system sizes; we find that there exists a remarkable correspondence in values calculated for a defined ratio of walklengths for the first two generations of this fractal set.