# A block preconditioner for non-isothermal flow in porous media

## TL;DR

This paper introduces a novel block preconditioner tailored for non-isothermal flow in porous media, effectively incorporating thermal effects like heat diffusion into reservoir simulations.

## Contribution

It develops a new block preconditioner with an efficient Schur complement approximation specifically designed for non-isothermal flow models.

## Key findings

- The preconditioner is scalable with problem size and parallelization.
- It effectively captures thermal effects such as heat diffusion.
- The solver improves convergence in thermal reservoir simulations.

## Abstract

In petroleum reservoir simulation, the industry standard preconditioner, the constrained pressure residual method (CPR), is a two-stage process which involves solving a restricted pressure system with Algebraic Multigrid (AMG). Initially designed for isothermal models, this approach is often used in the thermal case. However, it does not have a specific treatment of the additional energy conservation equation and temperature variable. We seek to develop preconditioners which better capture thermal effects such as heat diffusion. In order to study the effects of both pressure and temperature on fluid and heat flow, we consider a model of non-isothermal single phase flow through porous media. For this model, we develop a block preconditioner with an efficient Schur complement approximation. Both the pressure block and the approximate Schur complement are approximately inverted using an AMG V-cycle. The resulting solver is scalable with respect to problem size and parallelization.