Recently, a unified Eulerian-Lagrangian theory has been developed by one of us for nonreactive solute transport in space-time nonstationary velocity fields. We describe a combined analytical-numerical method of solution based on this theory for the special case of steady state flow in a mildly fluctuating, statistically homogeneous, lognormal hydraulic conductivity field. We take the unconditional mean velocity to be uniform but allow conditioning on measurements of log hydraulic conductivity (or transmissivity) and/or hydraulic head. This renders the velocity field nonstationary. We solve the conditional transport problem analytically at early time and express it in pseudo-Fickian form at later time. The deterministic pseudo-Fickian equations involve a conditional, space-time dependent dispersion tensor which we evaluate numerically along mean ''particle'' trajectories. These equations lend themselves to accurate solution by standard Galerkin finite elements on a relatively coarse grid. The final step is an explicit numerical computation of lower bounds on conditional concentration prediction variance-covariance (and coefficient of variation), travel time distribution, cumulative mass release across a ''compliance surface,'' the associated error, and plume spatial moments. Our method also allows quantification of the uncertainty in the original source location of any solute ''particle'' located anywhere in the field, at any time. This paper describes the methodology and presents some unconditional results. Conditioning and more advanced computations are presented in the subsequent papers. ¿ American Geophysical Union 1995