The 3D reconstruction of the pore space in Opalinus Clay is faced with the difficulty that high-resolution imaging methods reach their limits at the nanometer-sized pores in this material. Until now it has not been possible to image the whole pore space with pore sizes that span two orders of magnitude. Therefore, it has not been possible to predict the transport properties of this material with the help computer simulations that require 3D pore structures as input. Following the concept of self-similarity, a digital pore microstructure was constructed from a real but incomplete pore microstructure. The constructed pore structure has the same pore size spectrum as measured in the laboratory. Computer simulations were used to predict capillary pressure curves during drainage, which also agree with laboratory data. It is predicted, that two-phase transport properties such as the evolution of effective permeability as well as capillary pressures during drainage depend both on transport directions, which should be considered for Opalinus Clay when assessing its suitability as host rock for nuclear waste. This directional dependence is controlled on the pore scale by a geometric anisotropy in the pore space.