In this presentation, we introduce a novel numerical stabilization focused on mineral flow deposition in highly heterogeneous and anisotropic porous media. Our contribution involves interpreting an stabilized conforming finite element method based on residual minimization as a variational multiscale approach. We construct a coarse-scale approximation within continuous spaces by employing residual minimization on a dual Galerkin norm. Subsequently, we reinterpret the residual projection within the variational multiscale framework to derive a fine-scale approximation. This approach enables us to obtain both stable coarse- and fine-scale solutions, along with an on-the-fly error estimator to guide adaptivity. We demonstrate the effectiveness and reliability of our method through challenging scenarios involving advective-dominated flow in both linear and nonlinear numerical experiments, showcasing its performance in providing solution stability with an optimal convergence rate.