  We consider a toy model for a damped water waves system in a domain $\Omega_t
\subset \mathbb{T} \times \mathbb{R}$. The toy model is based on the
paradifferential water waves equation derived in the work of
Alazard-Burq-Zuily. The form of damping we utilize we utilize is a modified
sponge layer proposed for the three-dimensional water waves system by Clamond,
et. al. We show that, in the case of small Cauchy data, solutions to the toy
model exhibit a quadratic lifespan. This is done via proving energy estimates
with the energy being constructed from appropriately chosen vector fields.
