In this paper we consider a mixed finite element discretization for the diffusion equation on special nonmatching distorted hexahedral meshes. The model problem is motivated by applications in geosciences. We assume that the computational domain is presented as the union of two subdomains separated by, generally speaking, a nonplanar surface, the so-called fault interface. Each subdomain is divided into several `horizontal' layers corresponding to the media with different material properties. We construct logically cubic hexahedral meshes in each subdomain, which are conforming on the interfaces between different layers but are not obliged to match on the fault interface. We perform a special fault interface reconstruction algorithm in order to construct a conforming polyhedral mesh in the whole domain. Finally, we discretize the problem by the mixed finite element method invented in . The numerical results demonstrate good accuracy of the proposed method.