Both words can be used about using up time. The main difference is the familiarity. "spend" is a very common word, while "expend" is uncommon. On the Macmillan dictionary, "spend" shows three red stars (the most common) while "expend" shows none. (http://www.macmillandictionary.com/dictionary/british/spend_1) All this means you should use "spend".

In casual conversation, I would almost always use "spend" to talk about time. Expend can be used to describe the amount of time and effort one gives, but I think it's rarely used. I could say, "I spent a lot of time on it." Or I could say, "It was something in which I expended much of my time." "Expend" may sound awkward in normal conversation, so I would use "spend."

If i'm not wrong, you can use SPEND when you are talking about the time. http://dictionary.cambridge.org/dictionary/english/spend