The board, at the time of fixing the replenishment assessment rate, may provide by resolution that any producer operating a water-producing facility having a discharge opening not greater than two inches in diameter and providing ground water for domestic or irrigation uses on an area not exceeding one acre in extent, shall pay the amount fixed in such resolution as the replenishment assessment to be paid by such producer. No sworn statement as to the production of ground water from such water-producing facility need be filed.