A method for lubricating a turbomachinery system comprising a turbomachinery bearing, the method comprising: retaining a lubrication fluid with a gravity rundown lubrication tank disposed vertically above a main lubrication fluid supply header of a turbomachinery system, the main lubrication fluid supply header fluidly coupled to the turbomachinery bearing, the gravity rundown lubrication tank fluidly connected to the main lubrication fluid supply header by a flow control system configured to selectively flow lubrication fluid in response to a pressure differential between a lubrication fluid pressure in the main lubrication fluid supply header and a static pressure of the gravity rundown lubrication tank; in response to the lubrication fluid pressure in the main lubrication fluid supply header dropping below a backup fluid pressure threshold, flowing a first portion of the lubrication fluid from the gravity rundown lubrication tank to the main lubrication fluid supply header at a first pressure between the backup fluid pressure threshold and a shutdown trip pressure threshold of the main lubrication fluid supply header, the shutdown trip pressure threshold being less than the backup fluid pressure threshold, wherein the gravity rundown lubrication tank is raised to a vertical height such that the topmost level of the lubrication fluid is at a first vertical height determined to provide an initial static pressure equal to the first pressure; and in response to the lubrication fluid pressure in the main lubrication fluid supply header dropping below the shutdown trip pressure threshold, flowing a second portion of the lubrication fluid from the gravity rundown lubrication tank to the main lubrication fluid supply header at a second pressure less than or equal to the shutdown trip pressure threshold. retaining a lubrication fluid with a gravity rundown lubrication tank disposed vertically above a main lubrication fluid supply header of a turbomachinery system, the main lubrication fluid supply header fluidly coupled to the turbomachinery bearing, the gravity rundown lubrication tank fluidly connected to the main lubrication fluid supply header by a flow control system configured to selectively flow lubrication fluid in response to a pressure differential between a lubrication fluid pressure in the main lubrication fluid supply header and a static pressure of the gravity rundown lubrication tank; in response to the lubrication fluid pressure in the main lubrication fluid supply header dropping below a backup fluid pressure threshold, flowing a first portion of the lubrication fluid from the gravity rundown lubrication tank to the main lubrication fluid supply header at a first pressure between the backup fluid pressure threshold and a shutdown trip pressure threshold of the main lubrication fluid supply header, the shutdown trip pressure threshold being less than the backup fluid pressure threshold, wherein the gravity rundown lubrication tank is raised to a vertical height such that the topmost level of the lubrication fluid is at a first vertical height determined to provide an initial static pressure equal to the first pressure; and in response to the lubrication fluid pressure in the main lubrication fluid supply header dropping below the shutdown trip pressure threshold, flowing a second portion of the lubrication fluid from the gravity rundown lubrication tank to the main lubrication fluid supply header at a second pressure less than or equal to the shutdown trip pressure threshold. <EOS>