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Patterns and connectivity in random, heterogeneous or composite media affect many physical and geologic flow and transport phenomena. In physics, so-called percolation properties have been studied mostly on uncorrelated random fields. Yet, environmental and many engineered systems are inherently correlated in space. A key measure of connectivity in these system is the so-called percolation threshold, pc, which is known to be 31.16% in uncorrelated media. Our work shows that the percolation threshold in correlated random fields (e.g., aquifers) is only 12.6%, significantly lower than in the uncorrelated random fields. The percolation thresholds are similar for Markov chain, sequential Gaussian, and indicator random fields. The threshold is also a function of the system-size (finite-size effects) and of the correlation scale (relative to the resolution of the random field), decreasing with increasing correlation scale. Appropriate grid resolution and choice of simulation boundaries are critical to properly simulate connectivity in correlated natural geologic systems, which often have a finite extend. We then examine flow and transport properties in heterogeneous media. We derive a general solution for effective conductivity for an idealized, periodic heterogeneous media with cuboid inclusions. Comparison to effective conductivities derived for random heterogeneous media demonstrate similarities and differences in the behavior of the effective conductivity in regular periodic (low entropy) versus random (high entropy) media. The results define the low entropy bounds of effective conductivity in natural media, which is neither completely random nor completely periodic, over a large range of structural geometries. For isotropic inclusion and isoprobable conditions well below the percolation threshold, the results are in agreement with the self-consistent approach. For anisotropic cuboid inclusions, or at relatively close spacing in at least one direction (aniso-probable conditions), the effective conductivity of the periodic media is significantly different from that found in anisotropic random binary or Gaussian media. Transport properties in highly anisotropic systems with flow perpendicular to the anisotropy axis solute/particle transport in Markov Chain random fields is shown to be significantly different from Gaussian media, particularly near the percolation threshold.