The purpose of this paper is to provide the solvability and uniqueness result to the mixed Dirichlet–Robin boundary value problem for the nonlinear Darcy–Forchheimer–Brinkman system in a bounded, two-dimensional Lipschitz domain. First we obtain a well-posedness result for the linear Brinkman system with Dirichlet–Neumann boundary conditions, by reducing the problem to the system of boundary integral equations based on the fundamental solution of the Brinkman system and by analyzing this system employing a variational approach. The result is extended afterwards to the Poisson problem for the Brinkman system and to Dirichlet–Robin boundary conditions, using the Newtonian potential and the linearity of the solution operator. Further, we study the nonlinear Darcy–Forchheimer–Brinkman boundary value problem of Dirichlet and Robin type.