A block preconditioner for non-isothermal flow in porous media

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.

An efficient second-order energy stable BDF scheme for the space fractional Cahn-Hilliard equation

“…For large α (i.e., α = 1.9), the behaviour of our preconditioner is less satisfactory. Another more efficient preconditioners should be considered in our future work, such as the Sine transform-based preconditioner [21] and the two-stage preconditioner [40]. Furthermore, our method can easily be extended to the variable time-step version [41] and to the higher dimensional case.…”

Preconditioners for Krylov subspace methods: An overview

“…On the other hand, the size of A makes forming S and applying  u or  l too costly in general, and so many practical preconditioners are based on approximating A and S [128,[155][156][157]. We stress that this block triangular approach can be applied to any saddle point problem, for example porous media flow [158,159].…”

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

“…For the oil density ρ o , we choose a function of pressure and temperature given by an exponential correlation. Details and choices of parameters for both quantities can be found in Section 2.1.4 of [11].…”

“…Instead of solving a restricted pressure system in the first stage of CPR, we solve a restricted pressure-temperature system, resulting in a Constrained Pressure-Temperature Residual method (CPTR). In [11], we presented a Schur complement approximation for the pressure-temperature system for non-isothermal single phase flow. Such a Schur complement approximation leads to an effective block preconditioner.…”