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Numerical stability analysis of two‐dimensional solute transport along a discrete fracture in a porous rock matrix

This work reports numerical stability conditions in two‐dimensional solute transport simulations including discrete fractures surrounded by an impermeable rock matrix. We use an advective‐dispersive problem described in Tang et al. (1981) and examine the stability of the Crank‐Nicolson Galerkin finite element method (CN‐GFEM). The stability conditions are analyzed in terms of the spatial discretization length perpendicular to the fracture, the flow velocity, the diffusion coefficient, the matrix porosity, the fracture aperture, and the fracture longitudinal dispersivity. In addition, we verify applicability of the recently developed finite element method‐flux corrected transport (FEM‐FCT) method by Kuzmin ( ) to suppress oscillations in the hybrid system, with a comparison to the commonly utilized Streamline Upwinding/Petrov‐Galerkin (SUPG) method. Major findings of this study are (1) the mesh von Neumann number (Fo) ≥ 0.373 must be satisfied to avoid undershooting in the matrix, (2) in addition to an upper bound, the Courant number also has a lower bound in the fracture in cases of low dispersivity, and (3) the FEM‐FCT method can effectively suppress the oscillations in both the fracture and the matrix. The results imply that, in cases of low dispersivity, prerefinement of a numerical mesh is not sufficient to avoid the instability in the hybrid system if a problem involves evolutionary flow fields and dynamic material parameters. Applying the FEM‐FCT method to such problems is recommended if negative concentrations cannot be tolerated and computing time is not a strong issue.