This paper extends the work in Tabak and Cajueiro [Are the crude oil markets becoming weakly efficient over time, Energy Economics 29 (2007) 28-36] and Alvarez-Ramirez et al. [Short-term predictability of crude oil markets: a detrended fluctuation analysis approach, Energy Economics 30 (2008) 2645-2656]. In this paper, we test for the efficiency of WTI crude oil market through observing the dynamic of local Hurst exponents employing the method of rolling window based on multiscale detrended fluctuation analysis. Empirical results show that short-term, medium-term and long-term behaviors were generally turning into efficient behavior over time. However, in this way, the results also show that the market did not evolve along stable conditions for long times. Multiscale analysis is also implemented based on multifractal detrended fluctuation analysis. We found that the small fluctuations of WTI crude oil market were persistent; however, the large fluctuations had high instability, both in the short- and long-terms. Our discussion is also extended by incorporating arguments from the crude oil market structure for explaining the different correlation dynamics.