The interaction of free electrons with intense laser beams in vacuum is studied using a 3D test particle simulation model that solves the relativistic Newton-Lorentz equations of motion in analytically specified laser fields. Recently, a group of solutions was found for very intense laser fields that show interesting and unusual characteristics. In particular, it was found that an electron can be captured within the high-intensity laser region, rather than expelled from it, and the captured electron can be accelerated to GeV energies with acceleration gradients on the order of tens of GeV/cm. This phenomenon is termed the capture and acceleration scenario (CAS) and is studied in detail in this paper. The maximum net energy exchange by the CAS mechanism is found to be approximately proportional to a 2_o, in the regime where a_o > 100, where a_o = eE_o/m_ewc is a dimensionless parameter specifying the magnitude of the laser field. The accelerated GeV electron bunch is a macro-pulse, with duration equal or less than that of the laser pulse, which is composed of many micro-pulses that are periodic at the laser frequency. The energy spectrum of the CAS electron bunch is presented. The dependence of the energy exchange in the CAS on various parameters, e.g., a 2_o (laser intensity), w_o (laser radius at focus), tao (laser pulse duration), b_o (the impact parameter), and theta_i (the injection angle with respect to the laser propagation direction), are explored in detail. A comparison with diverse theoretical models is also presented, including a classical model based on phase velocities and a quantum model based on nonlinear Compton scattering.