In this talk, I will present an overview of sparse representations and their applications in solving inverse modeling problems involving PDEs that describe multi-phase flow in heterogeneous porous media. The related PDE-constrained inverse problems are often formulated to infer spatially distributed material properties from dynamic response measurements at scattered source/sink locations. Reconstructing three-dimensional images of the material properties from limited response data leads to severely ill-posed inverse problems with several challenging requirements, including computational complexity, geologic plausibility, and proper level/form of parameterization. After a brief introduction to the problem and the related challenges, I will discuss how such sparse representations of spatially distributed material properties can facilitate the formulation and solution of such ill-posed inverse problems. Throughout the talk, I will present several numerical examples to highlight the important properties and advantages of sparse inversion formulations.