  We consider the following repulsive-productive chemotaxis model: Let $p\in
(1,2)$, find $u \geq 0$, the cell density, and $v \geq 0$, the chemical
concentration, satisfying \begin{equation}\label{C5:Am} \left\{ \begin{array}
[c]{lll} \partial_t u - \Delta u - \nabla\cdot (u\nabla v)=0 \ \ \mbox{in}\
\Omega,\ t>0,\\ \partial_t v - \Delta v + v = u^p \ \ \mbox{in}\ \Omega,\ t>0,
\end{array} \right. \end{equation} in a bounded domain $\Omega\subseteq
\mathbb{R}^d$, $d=2,3$. By using a regularization technique, we prove the
existence of solutions of this problem. Moreover, we propose three fully
discrete Finite Element (FE) nonlinear approximations, where the first one is
defined in the variables $(u,v)$, and the second and third ones by introducing
${\boldsymbol\sigma}=\nabla v$ as an auxiliary variable. We prove some
unconditional properties such as mass-conservation, energy-stability and
solvability of the schemes. Finally, we compare the behavior of the schemes
throughout several numerical simulations and give some conclusions.
