This paper presents the use of symmetry reduction method resulting in new exact solutions
for the groundwater flow and transport equation. It is assumed that the radionuclides are transported
by advection-diffusion in a single fracture and diffusion in the surrounding rock-matrix. The
application of one-parameter group reduces the number of independent variables, and consequently
the governing PDE of (1+2)-dimension reduces to set of ODEs which are solved analytically. This
enables us to present some new exact time-dependent solutions of the advection-diffusion equation.