In deepwater drilling, controlling pressure in the oil well is crucial, as excessive pressures in the drilled hole can result in blowouts, leading to disastrous events like BP’s Gulf of Mexico oil spill last year.

The deeper the well, the higher the pressure, and the higher the risks associated with tapping oil from wells.

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During drilling, when the pressure applied to balance the hydrocarbon pressure in a well is not great enough to overcome that exerted by gas and fluids in the rock formation drilled, water, gas, oil, or other formation fluid can enter the hole. This is a “gas kick,” which in worst-case scenarios can lead to blowouts.

New analysis of a mathematical model has applications to gas kicks in deep-water oil wells, said Steinar Evje, professor of applied mathematics, petroleum in the Department of Petroleum Technology (IPT) at the University of Stavanger. Evje wrote a paper on the subject.

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The use of math models is important to develop tools that can help simulate, and increase control in deep-water well operations.

“Various gas kick simulators are being developed for the purpose of studying well control aspects during exploratory and development drilling,” Evje said. “Simulators have become an important tool for the development of new, more efficient and safer drilling methods.”

Schematic of an oil rig.

Schematic of an oil rig.

“A simulator for drilling operations is composed of a set of nonlinear coupled partial differential equations that describe the simultaneous flow of hydrocarbons in a well. This mathematical model represents a ‘virtual laboratory’ where the finer mechanisms related to a number of different physical effects can be studied in detail,” Evje said.

The main challenge presented in models is the precise prediction of the pressure profile in addition to liquid/gas volumes and flow rates at various points along the oil well.

“This issue becomes even more critical as many drilling operations today involve long and deep wells with corresponding high pressures and high temperatures,” Evje said. Regions along the well open to crevices and deformities in the rock formations present specific challenges, as it is critical to maintain well pressure at these positions within certain limits. Thus, in the case of inflow of gas from surrounding rock formations, it would be important to safely transport this gas out of the well.

The starting point for Evje’s proposed mathematical model is a one-dimensional two-phase model, which can simulate unsteady, compressible liquid and gas flow in pipes and wells. Unlike previously analyzed models, in this gas-liquid model, the two phases may have unequal fluid velocity and a generalized term to jointly represent liquid and gas pressure.

This allows a model that can describe the ascent of a gas slug (conglomerate of high pressure gas bubbles) due to buoyancy forces in a vertical well. A gas-kick situation usually accompanies such a flow scenario.

In order to compute reliable solutions, it is crucial to have a model well defined mathematically. Mathematical methods can help derive upper and lower limits for various quantities like masses and fluid velocities, which provide insight into the parameters that are important for the control of these quantities. In addition, they allow proof of the existence of solutions for the model in a strict mathematical sense. In his paper, Evje said under certain assumptions, a solution exists.

There is an assumption conditions are isothermal, and relevant physical mechanisms are factored into the model, such as frictional forces, hydrostatic pressure, force of gravity, and compression and decompression of gas.

Such mathematical analysis is essential to optimize and evaluate drilling operations and well-control practices in order to minimize the possibility of oil well disasters, especially in deep-water wells.

“The possibility of blowout occurrences needs to be mitigated in order to avoid human casualties, financial losses, and finally but not least, environmental damage,” Evje said.

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