Uniform flow of mean velocity U takes place in a highly heterogeneous, isotropic, aquifer of lognormal conductivity distribution (variance σY2, integral scale I). A conservative solute is injected instantaneously over an area A0 at x = 0, normal to the mean flow, in a flux-proportional mode. Longitudinal spreading is caused by advection by the fluid velocity and is quantified with the aid of the mass flux μ(t, x) through fixed control planes at x. An equivalent macrodispersivity is defined in terms of the traveltime variance. The flow and transport are solved numerically in three realizations of the conductivity field with σY2 = 2, 4, 8, respectively. The medium is modeled by a collection of a large number N = 100,000 of spherical inclusions whose conductivities are drawn at random. Transport is simulated by tracking 40,000 particles originating at a large injection area (A0 ≃ 2000 I2) and for travel distance x ≤ 121 I. It is found that the mass flux has a highly skewed time distribution because of the late arrival of solute particles that are moving through low-conductivity blocks. The tail leads to large values of the equivalent macrodispersivity, which is highly dependent on cutoffs corresponding to the arrival of even 0.999 or 0.995 of the total mass. Furthermore, the tail is nonergodic, as it depends on the plume size. Transport appears to be anomalous in the considered interval, although by the central limit theorem it has to tend asymptotically to Fickianity and Gaussianity.