Mathematical Modeling of Fluid Flow through Channels behind Well Casing

Oil and gas leakage through channels in the annular space of oil and gas wells has long been a problem responsible for sustained casing pressure and environmental issues. This type of channels forms due to low efficiency of cement placement during well cementing processes, gravity segregation after cementing horizontal wells, formation fluid invasion during the setting of cement slurry, and corrosion of formation gas, such as CO2 and H2 S over the life of well service. Successful sealing of the channels requires a rigorous hydraulics model for simulating the friction of sealants in the channel. The challenge is from the irregular shape of cross sections of the channels. Owing to the nature of circular shapes of wellbore and casing, bow-shape cross sections are assumed to represent these irregular shapes of cross sections. An analytical model is presented to describe laminar flow in channels of bow-shaped cross sections. The model is validated by a comparison with a traditional rectangular slot model for narrow cross sections. Result of model analysis indicates that use of rectangular cross sections to approximate bow-shaped cross sections will under-estimate the pressure gradient in narrow cross sections and over-estimate the pressure gradient in wide cross sections. A case analysis of a cement squeezing operation shows that the newly developed hydraulics model for fluid flow in channels of bow-shaped cross sections is easy to use in engineering applications.