Advection-conduction equation (ACE) used to explain the combined conduction and convection mechanisms of heat transfer thorough porous media has received considerable attention with a wide range of applications in various disciplines. The present study concentrates on developing a generalized analytical solution to ACE through incorporation of Fourier series and a new dependent variable for the problem of heat transfer through porous media in one-dimensional finite spatial region assuming transient boundary conditions. Then, assuming that a fracture acts as the time-dependent boundary condition, an analytical solution is obtained for a typical fractured porous medium. Based on the analytical solution, heat transfer shape factor is also obtained, on which the effects of governing parameters of matrix block such as Peclet number are investigated throughout transient and pseudo-steady-state (PSS) periods for the first time. Moreover, a correlation is generated to estimate the PSS heat transfer shape factor in terms of dimensionless time and Peclet number for constant concentration and linearly ascending temperature boundary conditions. Additionally, the numerical finite difference and finite element methods were utilized to validate the analytical solution results. The results demonstrated that the obtained generalized analytical solution can be regarded as a reliably excellent and accurate mathematical lever to describe the heat transfer phenomenon through porous media subject to determine transient boundary conditions and also to verify numerical results.