For both isothermal and thermal petroleum reservoir simulation, the Constrained
Pressure Residual (CPR) method is the industry-standard preconditioner. This method is a twostage 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.