The Murray-Darling Basin Authority has been accused of withholding information about how it manages the nation's biggest river system as it resists disclosures to the Senate and a Royal Commission that gets underway on Monday.

The authority, along with the Turnbull government, applied last week to have an injunction preventing officials from giving evidence at the South Australian Royal Commission. Former authority staffers are understood to have been told by the commission to seek legal advice before becoming witnesses.

Rex Patrick, a South Australian Centre Alliance senator, who has issued several orders for the production of documents, said the authority had "not been forthcoming with information".

"It's been like pulling teeth - actually, more like major surgery for a wisdom tooth," Senator Patrick said.