Hydrodynamic Dispersion Coefficients in a Porous Medium with Parallel Fractures -

The results form the application of the method of homogenization for the transport of solute in a porous medium with microscale pores and the rock matrix are applied to examine the dispersion characteristics of a solute in a porous rock matrix with parallel fractures including the retardation in the rock matrix and the decay effects. By using the solutions to a boundary value problem the effective macroscale dispersion coefficients (longitudinal and transverse) are examined for various parameters such as the fracture porosity, the rock matrix porosity, and the retardation. The transverse dispersivity is essentially of diffusion type and shows minor variation with the fracture porosity and the matrix porosity. The longitudinal dispersivity increases with the flow intensity (Peclet number) and the retardation except for very low retardation coefficients. It is also shown that the longitudinal dispersivity decreases with the fracture porosity and the matrix porosity. INTRODUCTION The dispersion of solute matter in a rock medium is important from the viewpoints of the management and the safety regard of the underground disposal of the nuclear wastes. Rock formations are typically characterized by the existence of fracture networks in which the ground water is transported and the rock matrix which is composed of solid phase and immobile water. If a solute matter is released in a rock medium, it migrates with the flowing fluid in the fracture. The fluid flow is usually very slow and is in the laminar flow regime. A released solute matter experiences not only diffusion in the rock matrix but also hydrodynamic dispersion in the fracture caused by the non-uniform fluid velocity distribution (Taylor dispersion). The solute in the rock matrix diffuses through the immobile water and also is sorbed onto the solid phase. Under the local equilibrium condition the solute is partitioned between the solids and the immobile water. This is effectively accounted for by the retardation coefficient of the rock matrix. The solute in the fracture while being transported by the flowing fluid experiences dispersion and interacts with the immobile water in the rock matrix through diffusion on the interface between the fracture and the rock matrix. WM’07 Conference, February 25-March 1, 2007, Tucson, AZ Studies on the behavior of radionuclides in a porous rock medium are mostly focused on the analysis in the individual fracture with the interaction with the surrounding host rock matrix including the distribution of the solute concentration in the fracture and the neighboring rock matrix[1-9]. From the viewpoint of the management of the underground repository and the environmental concerns over the rock domain it is necessary to assess and examine the behavior of the released contaminant on a larger macroscale including the spreading characteristics in the rock media with fractures. In this study, the derivation of the macroscale governing equation for solute transport in a saturated porous rock medium is briefly mentioned. Also a boundary value problem is defined which can be used to determine the dispersion coefficients. The medium is composed of pore space (Ωƒ) and the rock matrix (Ωm) which is further composed of the solid phase and the immobile water. The theoretical frame for deriving the effective relations on the macroscale is the method of homogenization which is known to be highly effective in the treatment of the heterogeneous medium. An explicit expression for calculating the dispersion coefficients on the macroscale is given. The results are applied to the transport of a solute matter in a porous rock medium with parallel fractures of uniform spacing. Using the approximate solutions to the microcell boundary value problem the longitudinal and transverse dispersion coefficients are calculated for various choices of the fracture porosity, the rock matrix porosity, the retardation coefficients and the flow intensity. The effects of the parameters on the dispersion coefficients are examined. THE GOVERNING RELATIONS ON THE MICROSCALE The porous medium is assumed to be composed of the matrix (Ωm) in which the solid rock phase and immobile water exist and the fluid phase(Ωƒ) through which the pore water flows. It is assumed that the porous medium is saturated by water so that Ωƒ is occupied by water(assumed to be incompressible). The basic governing equations on the microscale are summarized as follows. In Ωƒ the fluid flow is governed by the conservation laws of mass and momentum and the transport of solute is governed by the masss conservation: where u and cƒ are the fluid velocity and the solute concentration( [M/L]) in the fluid, and ρƒ and μƒ are the density and absolute viscosity of the fluid. Also Dƒ and λ are the diffusivity and decay coefficient of the solute respectively. In Ωm , the solute mass conservation is governed by