In this paper, an unstructured grid generation algorithm is presented to produce two- and three-dimensional grids in porous media with networks of discrete fractures. The proposed grid generation algorithm considers underground contours map data to adapt unstructured grids to geological geometries. This allows construction of more realistic geometrical models. Sample two- and three-dimensional unstructured grids have been generated in complex porous media with seven intersecting fractures. Two-dimensional grids were generated within 0.17-2.37 s of CPU time. Generation of three-dimensional grids including grid quality improvement measures took 109 s of CPU time. This final grid contains 763 fracture cells and 49, 403 matrix cells. Angle distribution histograms of the three-dimensional grids show no skewed and flat angles within fracture and matrix cells. Two- and three-dimensional computational unstructured grids have been generated for geometries similar to published fractured porous media test cases. Incompressible and immiscible water-oil flow simulations were then obtained using these computational grids. Simulations gave identical results with published data which confirms the computational feature of the proposed unstructured grid generation algorithm.