Fluid flow in isotropic porous media is well understood and commonly simulated. However, there exists a class of porous materials that exhibit two porosity scales with a stark permeability contrast between the two scales. An example of such material is shale, a fine-grained sedimentary rock estimated to form from 44% to 56% of all sedimentary rocks on earth. Although this rock is extremely heterogeneous at multiple scales, it is commonly assumed to be comprised of a rock matrix exhibiting transverse isotropy due to the existence of bedding planes, and organics embedded in the matrix. Transverse isotropy in the rock matrix could impact not only the mechanical properties of the rock but also its fluid flow properties due to the presence of preferentially oriented micro-fractures in the rock matrix. Fluid could also flow through the nanopores of the organics, which are so small in size that Darcy's law typically does not hold. This paper describes a hydromechanical model for this type of material, focusing on the preferential flow patterns induced by double porosity, transverse isotropy, and non-Darcy flow. Whereas this study was inspired by shale properties, the discussion revolves around a generic material with similar properties. Numerical results elucidate the roles played by double porosity, transverse isotropy, and non-Darcy flow on the preferential flow patterns in such porous material.