In this paper, we introduce a new fingering function into the Fayers macroscopic model, in which its single parameter represents the solvent concentration just inside the leading edge of the fingers. The properties of this model are then examined and compared with the Todd and Longstaff empirical model by using high-resolution simulation (HRS) to obtain detailed fingers, first in 2D and subsequently in 3D. Both finite-difference and particle-tracking techniques have been used and found to give similar average characteristics for the flows. The important properties that a macroscopic/empirical model should reproduce reliably axe (1) fractional flow, (2) average concentration versus distance, (3) total mobility and (4) gravity/viscous interactions. Macroscopic/empirical model capabilities are examined for each of these in turn and optimum parameters found for rectilinear flows, first for non-gravity flows in Part I, and then for cases where gravity has stabilizing and destabilizing influences in Part II.