  In this work, we consider the two and three-dimensional stochastic convective
Brinkman-Forchheimer (2D and 3D SCBF) equations driven by irregular additive
white noise $$\mathrm{d}\boldsymbol{u}-[\mu
\Delta\boldsymbol{u}-(\boldsymbol{u}\cdot\nabla)\boldsymbol{u}-\alpha\boldsymbol{u}-\beta|\boldsymbol{u}|^{r-1}\boldsymbol{u}-\nabla
p]\mathrm{d} t=\boldsymbol{f}\mathrm{d} t+\mathrm{d}\mathrm{W},\
\nabla\cdot\boldsymbol{u}=0,$$ for $r\in[1,\infty),$ $\mu,\alpha,\beta>0$ in
unbounded domains (like Poincar\'e domains) $\mathcal{O}\subset\mathbb{R}^d$
($d=2,3$) where $\mathrm{W}(\cdot)$ is a Hilbert space valued Wiener process on
some given filtered probability space, and discuss the asymptotic behavior of
its solution. For $d=2$ with $r\in[1,\infty)$ and $d=3$ with $r\in[3,\infty)$
(for $d=r=3$ with $2\beta\mu\geq 1$), we first prove the existence and
uniqueness of a weak solution (in the analytic sense) satisfying the energy
equality for SCBF equations driven by an irregular additive white noise in
Poincar\'e domains by using a Faedo-Galerkin approximation technique. Since the
energy equality for SCBF equations is not immediate, we construct a sequence
which converges in Lebesgue and Sobolev spaces simultaneously and it helps us
to demonstrate the energy equality. Then, we establish the existence of random
attractors for the stochastic flow generated by the SCBF equations. One of the
technical difficulties connected with the irregular white noise is overcome
with the help of the corresponding Cameron-Martin space (or Reproducing Kernel
Hilbert space). Finally, we address the existence of a unique invariant measure
for 2D and 3D SCBF equations defined on Poincar\'e domains (bounded or
unbounded). Moreover, we provide a remark on the extension of the above
mentioned results to general unbounded domains also.

Probability