We seek to capture the behavior of a dense brine as it travels through porous media. While the traditional model for single-fluid phase flow through porous media is so well-established that it is near universal, it has several notable deficiencies that lead us to consider alternate formulations. We improve upon the classical model with a method that utilizes the laws of thermodynamics. This method, known as thermodynamically constrained averaging theory (TCAT), provides a basis for which we can accurately simulate transport and diffusion of a high-concentration solution through porous media. The TCAT model is nonlinear and is comprised of two PDEs as well as a number of closure relations. We begin by rewriting the PDEs as a system of partial differential-algebraic equations (PDAEs). We then use the method of lines to transform our PDAEs into a large system of ODEs. From there, we use a stiff temporal integrator to solve for the time derivative and arrive at a 1D numerical solution. As this model is both nonlinear and nonsmooth, we will discuss results and numerical difficulties.