Matrix Formulation for Simultaneous Calculations of Pressure and Temperature in Wells and Pipelines

Abstract

This paper presents a novel approach to calculate pressure and temperature profile in a pipe. An implicit formulation is implemented in this model using a combined system of the three conservation equations (mass, momentum, and energy), and using the change of enthalpy and pressure to calculate the change of temperature. The derivation of the equations for temperature and pressure profile calculation uses the traditional approach of applying the mass, momentum and energy balance equations to a control volume of the pipe. For a pipe, usually the pressure and temperature values at one end are known. Due to the non-linearity of balance equation, an iterative process is required to calculate the values at the other end of the pipe. In the model proposed in this paper the system of equations was arranged in a matrix and vector form, all pressure and temperature for all the nodes are calculated in one calculation step, different from the traditional approach. The proposed model is used with both Black oil table correlations and a fully compositional model for calculating P-T profiles. It can be applied over the entire inclination angle range from horizontal to vertical. Conventionally, marching algorithm is used to calculate pressure from first cell to second cell until the end of the pipe is reached. The final form of energy balance is expressed in terms of enthalpy; therefore, P-H flash is used to calculate temperature profile. The calculation process is faster as all of it is done in one step and more accurate than the traditional approach. Traditionally the coupling of the mechanical energy and heat balance equation, uses a marching algorithm to determine pressure and temperature profile in a pipe, using a nested pair of loops, usually the external one for temperature and an internal for pressure. The proposed method applies a novel matrix formulation using a vectorized procedure to determine simultaneously pressure and temperature using their natural connection the enthalpy through the balance equations. This method aims to determine pressure-temperature profiles in a fast and accurate way. Vector and matrices approach is a tool that improves the performance of a code and utilizing it for pipeline calculation is unique, it opens a door for also coupling with reservoir simulators in an integrated approach. Since most flow assurance problems i.e. paraffin, hydrate are related to pressure and temperature dependent, the proper calculation of the P-T profile is a must. The proposed model provides a fast and precise calculation organizing the problem in a structured manner.