  We present in detail a nonperturbative method for vortex liquid systems. This
method is based on the resummation of an infinite subset of Feynman diagrams,
the so-called parquet graphs, contributing to the four-point vertex function of
the Ginzburg-Landau model for a superconductor in a magnetic field. We derive a
set of coupled integral equations, the parquet equations, governing the
structure factor of the two-dimensional vortex liquid system with and without
random impurities and the three-dimensional system in the absence of disorder.
For the pure two-dimensional system, we simplify the parquet equations
considerably and obtain one simple equation for the structure factor. In two
dimensions, we solve the parquet equations numerically and find growing
translational order characterized by a length scale $R_c$ as the temperature is
lowered. The temperature dependence of $R_c$ is obtained in both pure and
weakly disordered cases. The effect of disorder appears as a smooth decrease of
$R_c$ as the strength of disorder increases.
