On the Homogenization of Geological Fissured Systems With Curved non-periodic Cracks

Abstract: We analyze the steady fluid flow in a porous medium containing a network of thin fissures i.e. width $\mathcal{O}(\epsilon)$, where all the cracks are generated by the rigid translation of a continuous piecewise $C^{1}$ functions in a fixed direction. The phenomenon is modeled in mixed variational formulation, using the stationary Darcy's law and setting coefficients of low resistance $\mathcal{O}(\epsilon)$ on the network. The singularities are removed performing asymptotic analysis as $\epsilon \rightarrow 0$ which yields an analogous system hosting only tangential flow in the fissures. Finally the fissures are collapsed into two dimensional manifolds.