If transaction fees came in at an even rate over time all at the exact same level then they work fine for security, acting similarly to fixed block rewards. Unfortunately that isn't how it works in the real world. There's a very well established day/night cycle with fees going to zero overnight and even longer gaps on weekends and holidays. If in the future Bitcoin is entirely dependent on fees for security (scheduled very strongly) and this pattern keeps up (overwhelmingly likely) then this is going to become a serious problem.

What's likely to happen is that at first there will simply be no or very few blocks mined overnight. There are likely to be some, as miners at first turn off their mining rigs completely overnight then adopt the more sophisticated strategy of waiting until there are enough fees in the mempool to warrant attempting to make a block and only then doing it. Unfortunately the gaming doesn't end there. Eventually the miners with lower costs of operation will figure out that they can collectively reorg the last hour (or some time period) of the day overnight and this will be profitable. That's likely to cause the miners with more expensive operations to stop attempting mining the last hour of the day preemptively. 

What happens after that I'm not sure.