Lawrence Berkeley National Laboratory

Exact analytical solutions are presented for solute transport in an unsaturated fracture and porous rock matrix. The problem includes advective transport in the fracture and rock matrix as well as advective and diffusive fracture-matrix exchange. Linear sorption in the fracture and matrix and radioactive decay are also treated. The solution is for steady, uniform transport velocities within the fracture and matrix, but allows for independent specification of each of the velocities. The problem is first solved in terms of the solute concentrations that result from an instantaneous point source. Superposition integrals are then used to derive the solute mass flux at a fixed downstream position from an instantaneous point source and for the solute concentrations that result from a continuous point source. Solutions are derived for cases with the solute source in the fracture and the solute source in the matrix. The analytical solutions are closed-form and are expressed in terms of algebraic functions, exponentials, and error functions. Comparisons between the analytical solutions and numerical simulations, as well as sensitivity studies, are presented. Increased sensitivity to cross-flow and solute source location is found for increasing Peclet number. The numerical solutions are found to compare well with the analytical solutions at lower Peclet numbers, but show greater deviation at higher Peclet numbers.