Thin flow element and the problem of ill-conditioning

The problem of ill-conditioning in anisotropic and heterogeneous flow regions using plane and axisymmetric conditions is studied. A new concept, conditioning ratio, which covers all the possible factors significantly affecting the problem of ill-conditioning (e.g. aspect ratio, heterogeneity and anisotropy) is proposed. It has been shown that there is an upper limit for this conditioning ratio, beyond which the stiffness matrix of the modelled region will be ill-conditioned. Using this limiting value and considering the fact that flow behaviour in a single fracture is, basically, independent of the value of permeability normal to the plane of the fracture, k₁, a criterion is established to calculate the value of k₁ for thin elements. By adopting this criterion the problem of ill-conditioning could be avoided, regardless of the value of the aspect ratio. The reliability of the proposed criterion is examined by using thin elements with very large aspect ratio (up to 10⁶) in modelling the pumping problem from a well fully intersecting a vertical fracture. The comparison of the numerical results with the available analytical solutions is found to be very satisfactory.