We review the results obtained in several experimental campaigns with the INRS high-power laser system and determine the X-ray emission scaling from synchrotron radiation produced during laser wakefield acceleration (LWFA) of electrons. The physical processes affecting the generation of intense and stable X-ray beams during the propagation phase of the high-intensity ultrashort pulse in the gas jet target are discussed. We successfully produced stable propagation in the gas jet target of a relativistic laser pulse through self-guiding on length larger than the dephasing and depletion lengths, generating very intense beams of hard X-rays with up to 200 TW on target. The experimental scaling law obtained for the photon yield in the 10-40 keV range is presented and the level of X-ray emission at the 1 PW laser peak power level, now available at several laser facilities, is estimated.