A general and efficient formulation of fractures and boundary conditions in the finite element method

The need to assess quantitatively the safety of waste repositories in deep geological media has fostered the development of e cient numerical models of groundwater ow and contaminant transport in fractured media. These models usually account for water ow through fracture zones embedded in a 3D rock matrix continuum. The rst formulation of fractures in groundwater ow nite element models was proposed by Kiraly, and later revisited and generalized by Perrochet. From a mathematical viewpoint, fractures can be considered as m-dimensional manifolds in an n-dimensional Euclidean space (m6n). The key step of this formulation lies in an expression relating the hypersurface element dSm to the in nitesimal local co-ordinates d i (i=1; : : : ; m). Here we present a novel proof for this relation using a di erent approach to that of Perrochet, and explore the e ciency and accuracy of the formulation. It is shown that the aforementioned relation leads to a general and compact formulation which is not only applicable to elements of any dimension (e.g. 1D, 2D and 3D elements in a 3D domain), but also overcomes the cumbersome and case-speci c calculations of traditional approaches. This formulation has been implemented in a versatile nite element program for modelling groundwater ow, solute transport and heat transport in porous and fractured media. The e ciency and accuracy of the proposed formulation has been analysed using a synthetic case dealing with ow and solute transport through a 2D fractured rock block. The proposed formulation, in which fractures are discretized by means of 1D elements is more e cient and accurate than the traditional nite element formulation of discretizing fractures by means of 2D elements. The capability of the proposed formulation to cope with complex systems is illustrated with a case study of groundwater ow induced by the construction of the access tunnel to an underground research laboratory in A spo (Sweden). The numerical model is able to reproduce the observed records of water levels in boreholes and ow rates into the tunnel. Although the proposed formulation has been implemented and tested within the framework of groundwater ow and solute transport in fractured porous media, it should be of interest for other boundary value problems