# A constrained pressure-temperature residual (CPTR) method for non-isothermal multiphase flow in porous media

## TL;DR

This paper introduces the CPTR method, an extension of the CPR preconditioner, which incorporates heat diffusion by solving a pressure-temperature system, improving thermal reservoir simulations.

## Contribution

The paper develops and evaluates a new CPTR preconditioner that extends CPR to include heat diffusion effects in thermal multiphase flow simulations.

## Key findings

- Enhanced preconditioning for thermal reservoir simulation.
- Improved convergence in multiphase flow cases.
- Effective for non-isothermal reservoir models.

## Abstract

For both isothermal and thermal petroleum reservoir simulation, the Constrained Pressure Residual (CPR) method is the industry-standard preconditioner. This method is a two-stage process involving the solution of a restricted pressure system. While initially designed for the isothermal case, CPR is also the standard for thermal cases. However, its treatment of the energy conservation equation does not incorporate heat diffusion, which is often dominant in thermal cases. In this paper, we present an extension of CPR: the Constrained Pressure-Temperature Residual (CPTR) method, where a restricted pressure-temperature system is solved in the first stage. In previous work, we introduced a block preconditioner with an efficient Schur complement approximation for a pressure-temperature system. Here, we extend this method for multiphase flow as the first stage of CPTR. The algorithmic performance of different two-stage preconditioners is evaluated for reservoir simulation test cases.