  We explore the role of matter effect in the evolution of neutrino oscillation
parameters in the presence of lepton-flavor-conserving and
lepton-flavor-violating neutral-current non-standard interactions (NSI) of the
neutrino. We derive simple approximate analytical expressions showing the
evolution/running of mass-mixing parameters in matter with energy in the
presence of standard interactions (SI) and SI+NSI (considering both positive
and negative values of real NSI parameters). We observe that only the NSI
parameters in the (2,3) block, namely $\varepsilon_{\mu\tau}$ and $(\gamma -
\beta) \equiv (\varepsilon_{\tau\tau} - \varepsilon_{\mu\mu})$ affect the
running of $\theta_{23}$. Though all the NSI parameters influence the evolution
of $\theta_{13}$, $\varepsilon_{e\mu}$ and $\varepsilon_{e\tau}$ show a
stronger impact at the energies relevant for DUNE. $\theta_{12}$ quickly
approaches to $\sim$ $90^{\circ}$ with increasing energy in both SI and SI+NSI
cases. The change in $\Delta m^2_{21,m}$ is quite significant as compared to
$\Delta m^2_{31,m}$ both in SI and SI+NSI frameworks. Flipping the signs of the
NSI parameters alters the way in which mass-mixing parameters run with energy.
We demonstrate the utility of our approach in addressing several important
features related to neutrino oscillation such as: a) unraveling interesting
degeneracies between $\theta_{23}$ and NSI parameters, b) estimating the
resonance energy in presence of NSI when $\theta_{13}$ in matter becomes
maximal, c) figuring out the required baselines and energies to have maximal
matter effect in $\nu_{\mu}$ $\rightarrow$ $\nu_{e}$ transition in the presence
of different NSI parameters, and d) studying the impact of NSI parameters
$\varepsilon_{\mu\tau}$ and $(\gamma - \beta)$ on the $\nu_{\mu} \to \nu_{\mu}$
survival probability.
