The binary system of hexaethylene glycol n-hexadecyl ether (C16EO6) and water ((H2O)-H-2) has a complex, temperature-dependent lyotropic phase sequence, in the concentration region of 48-62 wt %. On cooling it shows the sequence lamellar phase, L-alpha, random mesh phase Mh(1)(0), rhombohedral mesh phase, Mh(R (3) over bar hm), bicontinuous cubic phase, V-1(Ia (3) over bar hd), and a two-phase hexagonal region, H-1 + L-beta. On heating from the latter two- phase region the phase sequence is V-1(Ia (3) over bar hd), Mh(1)(0), and L alpha. Polarizing optical microscopy, H-2 nuclear magnetic resonance, and small-angle X-ray scattering have been used to study the stability of these phases, their sequence, and their physical parameters with the addition of the oils, 1-hexene, decane, and octadecane. The oils are located within the alkyl chain regions of the mesophase structures. Depending on whether the added oil is "penetrating" or "swelling", it may reside in the region between the C-16 alkyl chains of the surfactant or at the center of the bilayer and affect phase stability. Oils affect both the volume of the alkyl chain region (at fixed surfactant water mole ratio) and the rigidity of the interfacial region. Both effects can influence the phase structures and their ranges of stability. Adding different types of oil to the mesh phases gives an opportunity to understand the factors that are important in their formation. The transition from the Mh(1)(R (3) over barm) phase to Mh(1)(0) phase is triggered by the hydrocarbon region swelling to a critical volume fraction of 0.32, a surfactant rod radius of similar to 1.75 nm, and a critical water layer thickness of similar to 2.5 nm. The latter is most likely responsible for a weakening of the interlayer headgroup overlap interaction and the loss of correlation between the layers. The lamellar phase becomes the only stable phase at high oil content.