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The Neil Gehrels Swift observatory observes gamma-ray burst (GRB) plateaus in X-rays. We test the reliability of the closure relations through the fireball model when dealing with GRB plateau emissions. We analyze 455 X-ray light curves collected by Swift from 2005 January until 2019 August for which the redshifts are either known or unknown using the phenomenological Willingale 2007model. Using these fits, we analyze the emission mechanisms and astrophysical environments of these GRBs through the closure relations within the time interval of the plateau emission. Finally, we test the three-dimensional fundamental plane relation (Dainotti relation) which connects the prompt peak luminosity, the time at the end of the plateau (rest frame), and the luminosity at that time, for the GRBs with redshift, concerning groups determined by the closure relations. This allows us to check if the intrinsic scatter σintof any of these groups is reduced compared to previous literature. The most fulfilled environments for the electron spectral distribution, p> 2, are wind slow cooling (SC) and interstellar material (ISM) slow cooling for cases in which the parameter q, which indicates the flatness of the plateau emission and accounts for the energy injection, is 0 and 0.5, respectively, in cases with both known and unknown redshifts. We also find that for short GRBs all ISM environments with q= 0 have the smallest σint= 0.04 ± 0.15 in terms of the fundamental plane relation holding a probability of occurring by chance of p= 0.005. We have shown that the majority of GRBs presenting plateau emission fulfill the closure relations, including the energy injection, with a particular preference for the wind SC environment. The subsample of GRBs that fulfill the given relations can be used as possible standard candles and can suggest a way to reduce the intrinsic scatter of these studied relationships.