  By coupling the EUropean Heliospheric FORcasting Information Asset (EUHFORIA)
and the improved Particle Acceleration and Transport in the Heliosphere (iPATH)
model, two energetic storm particle (ESP) events, originating from the same
active region (AR 13088) and observed by Solar Orbiter (SolO) on August 31 2022
and September 05 2022, are modelled. While both events originated from the same
active region, they exhibited notable differences, including: 1) the August ESP
event lasted for 7 hours, while the September event persisted for 16 hours; 2)
The time intensity profiles for the September event showed a clear cross-over
upstream of the shock where the intensity of higher energy protons exceeds
those of lower energy protons, leading to positive (``reverse'') spectral
indices prior to the shock passage. For both events, our simulations replicate
the observed duration of the shock sheath, depending on the deceleration
history of the CME. Imposing different choices of escaping length scale, which
is related to the decay of upstream turbulence, the modelled time intensity
profiles prior to the shock arrival also agree with observations. In
particular, the cross-over of this time profile in the September event is well
reproduced. We show that a ``reverse'' upstream spectrum is the result of the
interplay between two length scales. One characterizes the decay of upstream
shock accelerated particles, which are controlled by the energy-dependent
diffusion coefficient, and the other characterizes the decay of upstream
turbulence power, which is related to the process of how streaming protons
upstream of the shock excite Alfv\'{e}n waves. Simulations taking into account
real-time background solar wind, the dynamics of the CME propagation, and
upstream turbulence at the shock front are necessary to thoroughly understand
the ESP phase of large SEP events.
