In two-dimensional bounded Lipschitz domains, we analyze a convective Brinkman-Forchheimer problem on the weighted spaces H01(omega,omega)xL2(omega,omega)/R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\textbf {H}}}_0<^>1(\omega ,\varOmega ) \times L<^>2(\omega ,\varOmega )/{\mathbb {R}}$$\end{document}, where omega\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega $$\end{document} belongs to the Muckenhoupt class A2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_2$$\end{document}. Under a suitable smallness assumption, we prove the existence and uniqueness of a solution. We propose a finite element method and obtain a quasi-best approximation result in the energy norm a la Cea under the assumption that omega\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varOmega $$\end{document} is convex. We also develop an a posteriori error estimator and study its reliability and efficiency properties. Finally, we develop an adaptive method that yields optimal experimental convergence rates for the numerical examples we perform.