This paper presents an analysis of the mathematical structure of three-component and four-component gas displacements. The structure of one-dimensional flows in which components partition between two phases is governed by the geometry of a set of equilibrium tie lines. We demonstrate that for systems of four components, the governing mass conservation laws for the displacement can be represented by an eigenvalue system whose coefficient matrix has a global triangular structure, which is defined in the paper, for only specific types of phase behavior. We show that four-component systems exhibit global triangular structure if and only if (1) tie lines meet at one edge of the quaternary phase diagram or (2) if tie lines lie in planes. For such systems, shock and rarefaction surfaces coincide and are planes. We prove that systems are of category (2) if equilibrium ratios (K-values) are independent of mixture composition. In particular, for such systems shock and rarefaction curves will coincide. We also show that for systems with variable K-values, the rarefaction surfaces are almost planar in a precise sense, which is described in the paper. Therefore, systems with variable K-values may be well approximated by assuming shock and rarefaction surfaces do coincide. For these special systems the construction of solutions for one-dimensional, two-phase flow with phase behavior simplifies considerably. In Part II, we describe an application of these ideas to systems in which K-values are constant.