Germany was the only European Great Power without vital interests in the Mediterranean. Of vital interest to the Russians was the grain flow out of the Black Sea. In a Franco-Russian vs CP war, Russia would expect to ship supplies to it's Med force through the Dardanelles with Turkey being neutral.
To a great degree the Russian naval buildup on the Black Sea was due to the Ottomans' planning purchase of 2 modern dreadnoughts so it was not going well with an idea of a neutral Ottoman Empire. OTOH, if the Ottoman Empire remained neutral then the problem with grain shipping did not require Russian naval presence on the Med where the French presence was already quite strong.
In November 1913 the navy called for a Dreadnought programme that would enable Russia to be able to have a force of 11 ships including Dreadnoughts and supporting ships in the Aegean by 1919. This was to support the Black Sea Fleet against the Turks.
So the Turks were not expected to be neutral after all?
The whole idea was approximately century too late and was a senseless repetition of the Russian Mediterranean expeditions of the XVIII and XIX centuries because supply problems for the modern ships were considerably more complicated than those of the age of sails and cannonballs and in the case of a hostile Ottoman Empire (and presumably hostile Germany as well), there would be very little chance for the Russian Med squadron to get back home or to get necessary supplies from home (Black Sea and Baltic Sea closed by the enemies leaving an option of carrying supplies either from Murmansk/Archangelsk (with a risk of being intercepted by the Germans) or all the way from Vladivostok. France, as WWI proved, was rather unreliable partner in the terms of supplies (memoirs of the Russian military attache in France during WWI depict quite depressing picture) or even usage of the Russian funds allocated in France (I already provided some data in an earlier post).
All these grand schemas had been a byproduct of 2 main factors: general obsession of the navies with Mahan's ideas (without any serious attempt to adjust them to the national specifics and interests) and idiotic ideas regarding the foreign policy propagated by the "Prince of the Straits", Izvolsky. Army's position that the main purpose of the Russian navy should be to help defending the coasts was solid and its implementation would be much less costly allowing to spend more money on army's rearmament (and perhaps building up domestic weapons production), which proved to be a critical issue during WWI while both Baltic and Black Sea fleets proved a marginal factor. Russia had limited financial resources and an ambitious naval program was a luxury it could ill afford.