Evaluation of bottomhole pressure from measured surface pressure during an acid stimulation treatment provides necessary information to determine the evolution of skin factor as the treatment proceeds. Calculation of bottomhole pressure could be very complicated when compressible fluids are involved in the injection process, such as when foam is used as a diverting agent.
This paper presents a mathematical model for predicting bottomhole pressure from measured surface pressure. With the aid of the continuity equation and the equation of state, we developed a unique solution to the steady-state mechanical energy-balance equation. The density variation is included through an equation of state (EOS). Viscosity variation is considered through the Reidenbach's viscosity model, which is a laminar flow model of a yield pseudoplastic fluid with viscosity dependent on foam quality and fluid rheological properties. For different stimulation fluids injected into the wellbore, the length of each fluid column has to be tracked as the treatment proceeds, and the reduction of column length for compressible fluid as it flows down the wellbore is considered for accurate bottomhole estimation.
The model was tested against previously published data with satisfactory results. We illustrate in this paper the applications of the model by showing the complex surface pressure behavior that can occur when foams are used in acid treatments. The model is flexible and reliable, and can easily be adapted to a skin factor calculation model in matrix stimulation for stimulation treatment evaluation.