  This study delves into a comprehensive examination of the three-dimensional
$(3D)$ incompressible magneto-hydrodynamic $(MHD)$ equations in
$H^{1}(\R^{3})$. The modification involves incorporating a power term in the
nonlinear convection component, a particularly relevant adjustment in porous
media scenarios, especially when the fluid adheres to the Darcy-Forchheimer law
instead of the conventional Darcy law. Our main contributions include
establishing global existence over time and demonstrating the uniqueness of
solutions. It is important to note that these achievements are obtained with
smallness conditions on the initial data, but under the condition that $\beta
>3$ and $\alpha>0$. However, when $\beta=3$, the problem is limited to the case
$0<\alpha<\frac{1}{2}$ as the above inequality is unsolvable for these values
of $\alpha$ using our method. To support our statement, we will add a "slight
disturbance" of the function f of the type $f(z)=log(e+z^{2})$ or
$\log(\log(e^{e}+z^{2}))$ or even $\log(\log(\log((e^{e})^{e}+z^{2})))$.
