In this series of two papers, we present a front-tracking method for the numerical simulation of first-contact miscible gas injection processes. The method is developed for constructing very accurate (or even exact) solutions to one-dimensional initial-boundary-value problems in the form of a set of evolving discontinuities. The evolution of the discontinuities is given by analytical solutions to Riemann problems. In this paper, we present the mathematical model of the problem and the complete Riemann solver, that is, the analytical solution to the one-dimensional problem with piecewise constant initial data separated by a single discontinuity, for any left and right states. The Riemann solver presented here is the building block for the front-tracking/streamline method described and applied in the second paper.