Australia's richest person Gina Rinehart has lost an application for a new suppression order on her family trust court case.

After an urgent hearing in the High Court on Wednesday, Ms Rinehart won the right to keep details of legal action taken against her by three of her children secret until March 9.

But late on Thursday afternoon her lawyers lodged an application in the New South Wales Supreme Court seeking a permanent suppression order on the new ground of "safety fears" for herself, her children and grandchildren if the case is reported.

The court was given an "expert" report on security that said members of the billionaire's family would be at risk from foreign governments and terrorists if the suppression order was lifted.

Justice Michael Ball ruled against the application.

He said Ms Rinehart's lawyer submitted that if the case received publicity it was "likely, or at least it may, attract the attention of persons who could constitute a threat to the parties to the litigation".

"There can be no doubt that there is considerable media interest in this litigant," Justice Ball said.

He also agreed that publicity of the proceedings might attract unwelcome comment or attention to the parties.

"But I think the same is true of any of the conduct that those parties may engage in."

Unwelcome comment or attention was not sufficient to establish the necessity to protect the litigants' safety by a suppression order, the judge said.

"It seems to me, if it were otherwise, a suppression order would be justified on that ground in respect of any proceedings involving a person with a significant public profile," he said.

Justice Ball said an affidavit by Ms Rinehart's lawyer attached material which was available in the public domain about the parties.

It also attached correspondence between them expressing concerns about their personal security in regard to publicity the case had attracted.

But the judge said he did not see how suppression of this material was necessary to protect their safety.

He noted the three children suing their mother did not want the suppression orders, even though they were said to be related to their safety.

'Safety risks'

The court was given a 44-page "expert" report on security prepared by consultancy Control Risks Australia Pacific.

It said members of the Rinehart family would be at risk from organised crime, petty criminals, deranged persons, maleficent employees or business partners, competitors, issue-motivated groups, foreign governments and terrorists if the secrecy order was lifted.

The report listed some case studies where increased media reporting has led to security threats.

The case studies are of soccer player David Beckham and his family, English singer Joss Stone, American television presenter David Letterman, Mexican football player El Gato, The Pulver family and Australian cardiac surgeon Victor Chang.

In an email revealed in court between Ms Rinehart and her eldest daughter Hope Rinehart Welker, the latter said "I don't think you understand what it means now that the whole world thinks you're going to be wealthier than Bill Gates - it means we all need bodyguards and very safe homes."

Three of Ms Rinehart's four children are involved in a dispute with her over a trust set up by their grandfather Lang Hancock before he died.

The proceedings against Ms Rinehart have been brought by Ms Rinehart Welker, John Langley Hancock and Bianca Hope Rinehart.

A fourth child, Ginia Rinehart, has sided with her mother.

Ms Rinehart is Australia's richest person with a personal wealth estimated to top $10.3 billion.

This week she upped her ownership of Fairfax Media, the publisher of The Age and the Sydney Morning Herald, from 5 per cent to a reported 13 per cent.

ABC/AAP