A Mathematical Model of a Gas Kick
Authors: Leblanc J.L. , Lewis R.L. 
Published: December 2007

Abstract
This study presents an analysis of annular backpressure variations associated with controlled gas kicks and their pronounced erect on casing strings and exposed underlying formations.
A mathematical model describing the volumetric behavior of an extraneous gas as it is transported from reservoir to surface conditions under changing temperatures and pressures has been programmed in a Kingston FORTRAN II language for digital computer analysis. The gases under investigation typify Gulf Coast reservoir gases within a 0.6 to 0.7 specific gravity range. The program output has been substantiated by actual field cases of gas kicks encountered in Gulf Coast wells.
The development of empirical equations for calculating suitable gas deviation factors for unique temperatures and pressures was incorporated in the program to provide realistic solutions. An output listing of annular backpressures and corresponding equivalent fluid densities resulting at a predetermined critical depth (casing setting depth) and total depth for selected stages of circulation is provided in a chronological sequence. Additional information including reservoir pressure and temperature, kill mud density, produced gas or surface volume of the expanded gas intrusion, drill pipe and annular volumes can be obtained from the model.
This paper illustrates that a precise knowledge of the volumetric behavior of extraneous gases in annular flow and its effect on equivalent fluid densities at a critical depth is significant and should receive serious consideration in controlling threatened blowouts and in the design of drilling programs. Surface pressures in excess of formation limitations are a threat to zones of lost circulation and are potentially injurious to productive intervals. A knowledge of annular backpressure and equivalent fluid density profiles for probable gas kicks aids in a technological accomplishment of drilling programs and provides a sale tolerance in the event a threatened blowout is encountered.

Introduction
Drilling operations are frequently interrupted when the drill bit penetrates permeable gas sands with reservoir pressures greater than that exerted by the drilling fluid. The differential pressures resulting permit an extraneous influx of gas into the wellbore. A suspension in drilling progress is necessary to restore fluid equilibrium throughout the system. Whether formation gas kicks originate unintentionally or by design, the prospect of a threatened or actual blowout exists and a method assuring a safe and effective well control procedure must be observed.
A significant contribution to well control technology was advanced by Records et al. in 1962. Using the concept of transmitting a constant equivalent formation pressure at the point of intrusion, Records et al. introduced a calculation technique providing the annular backpressures encountered in a well control environment as a function of the volumetric behavior of a 0.6 specific gravity natural gas. In essence, the procedure outlined an annular backpressure schedule in terms of fluid volume circulated at different stages of a well control operation.
A number of other publications proposing various techniques for controlling gas intrusions in a wellbore achieve pressure control essentially through maintenance of a constant bottom-hole pressure by surface choke adjustments. The subsequent pressure effects induced in the annulus unfortunately receive little emphasis.
Due to the tedious and repetitive nature of annular backpressure computations, a theoretical solution by digital computer is introduced for predicting annular backpressure and equivalent fluid density profiles associated with controlled gas kicks. We point out the effects of volumetric behavior of extraneous gases in annular flow and related field phenomena on equivalent fluid densities at a critical depth. The investigation indicated that equivalent fluid densities at a critical depth are of significance and should receive consideration in the control of threatened blowouts and in the design of drilling programs.