As a therapeutic strategy for neurodegenerative diseases and brain cancers, convection-enhanced drug delivery (CED) directly infuses a large molecular weight of drugs into target cells. Despite the success of many previous in-vitro experiments on CED, challenges still remain, including catheter backflow, limited dispersion, and adapting delivery vectors to human anatomy and immunologic systems. In particular, a mathematically-based predictive model should be able to provide the interstitial spread, concentration, and pressure of the infusate at the level that suffices to form a basis for treatment planning. In this study, we aim to suggest a simple but rigorous approach that can properly capture the convective (advective) and diffusive transport of infusate via well-controlled injection tests. For this purpose, we investigate the advection-dispersion transport of an infusate (bromophenol blue solution) in the brain surrogate (agarose gel 0.2% w/w) at different injection rates, ranging from 0.25 to 4 ?l/min, by closely monitoring changes in the color intensity, propagation distance, and injection pressures. One of the salient observations is that the flow pattern is spherical (i.e., uniform) for injection rates up to 2 µl/min, while a rather disk-shaped (i.e., localized) propagation of the infusate becomes prevalent at the injection rate of 4 µl/min. Analytical estimates using the 1-D closed-form solution with the coefficient of molecular diffusion and average velocity values show a good agreement with the experimental data for the infusion volume of less than 100 ?l for the uniform flow pattern. However, the analytical estimates begin to underestimate the infusate propagation as the injection continued. The seepage velocity, which can be obtained by curve fitting with experimental data, is shown to be greater than the average velocity to some extent, particularly for the later infusion time elapses. The poroelastic deformation in the brain surrogate may lead to changes in porosity, and consequently, slight increases in the actual flow velocity as infusion continues. Lastly, the obtained infusion behaviors are analyzed using the dimensionless propagation curves and Péclet number, and we discuss a potential strategy to improve the efficacy of CED.