  Based on a decomposition of the magnetic field into potential and
nonpotential components, magnetic energy and relative helicity can both also be
decomposed into two quantities: potential and free energies, and
volume-threading and current-carrying helicities. In this study, we perform a
coupled analysis of their behaviors in a set of parametric 3D
magnetohydrodynamic (MHD) simulations of solar-like eruptions. We present the
general formulations for the time-varying components of energy and helicity in
resistive MHD. We calculated them numerically with a specific gauge, and
compared their behaviors in the numerical simulations, which differ from one
another by their imposed boundary-driving motions. Thus, we investigated the
impact of different active regions surface flows on the development of the
energy and helicity-related quantities. Despite general similarities in their
overall behaviors, helicities and energies display different evolutions that
cannot be explained in a unique framework. While the energy fluxes are similar
in all simulations, the physical mechanisms that govern the evolution of the
helicities are markedly distinct from one simulation to another: the evolution
of volume-threading helicity can be governed by boundary fluxes or helicity
transfer, depending on the simulation. The eruption takes place for the same
value of the ratio of the current-carrying helicity to the total helicity in
all simulations. However, our study highlights that this threshold can be
reached in different ways, with different helicity-related processes dominating
for different photospheric flows. This means that the details of the
pre-eruptive dynamics do not influence the eruption-onset helicity-related
threshold. Nevertheless, the helicity-flux dynamics may be more or less
efficient in changing the time required to reach the onset of the eruption.
