# Algebraic Multigrid Preconditioners for Multiphase Flow in Porous Media

## Abstract

Multiphase flow is a critical process in a wide range of applications, including carbon sequestration, contaminant remediation, and groundwater management. Typically, this process is modeled by a nonlinear system of partial differential equations derived by considering the mass conservation of each phase (e.g., oil, water), along with constitutive laws for the relationship of phase velocity to phase pressure. In this study, we develop and study efficient solution algorithms for solving the algebraic systems of equations derived from a fully coupled and time-implicit treatment of models of multiphase flow. We explore the performance of several preconditioners based on algebraic multigrid (AMG) for solving the linearized problem, including “black-box” AMG applied directly to the system, a new version of constrained pressure residual multigrid (CPR-AMG) preconditioning, and a new preconditioner derived using an approximate Schur complement arising from the block factorization of the Jacobian. Finally, we show that the new methods are the most robust with respect to problem character, as determined by varying effects of capillary pressures, and we show that the block factorization preconditioner both is efficient and scales optimally with problem size.

- Authors:

- University of Maryland; Univ. of Maryland, College Park, MD (United States). Applied Math, Stats, and Scientic Computation
- Univ. of Maryland, College Park, MD (United States). Dept. of Computer Science
- Los Alamos National Lab. (LANL), Los Alamos, NM (United States)

- Publication Date:

- Research Org.:
- Los Alamos National Lab. (LANL), Los Alamos, NM (United States)

- Sponsoring Org.:
- USDOE Office of Environmental Management (EM); National Science Foundation (NSF)

- OSTI Identifier:
- 1505960

- Report Number(s):
- [LA-UR-16-28063]

[Journal ID: ISSN 1064-8275]

- Grant/Contract Number:
- [89233218CNA000001; SC0009301; DMS1418754; AC52-06NA25396]

- Resource Type:
- Accepted Manuscript

- Journal Name:
- SIAM Journal on Scientific Computing

- Additional Journal Information:
- [ Journal Volume: 39; Journal Issue: 5]; Journal ID: ISSN 1064-8275

- Publisher:
- SIAM

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICS AND COMPUTING; Mathematics; Multiphase flow; preconditioning; multigrid

### Citation Formats

```
Bui, Quan, Elman, Howard, and Moulton, John David. Algebraic Multigrid Preconditioners for Multiphase Flow in Porous Media. United States: N. p., 2017.
Web. doi:10.1137/16M1082652.
```

```
Bui, Quan, Elman, Howard, & Moulton, John David. Algebraic Multigrid Preconditioners for Multiphase Flow in Porous Media. United States. doi:10.1137/16M1082652.
```

```
Bui, Quan, Elman, Howard, and Moulton, John David. Thu .
"Algebraic Multigrid Preconditioners for Multiphase Flow in Porous Media". United States. doi:10.1137/16M1082652. https://www.osti.gov/servlets/purl/1505960.
```

```
@article{osti_1505960,
```

title = {Algebraic Multigrid Preconditioners for Multiphase Flow in Porous Media},

author = {Bui, Quan and Elman, Howard and Moulton, John David},

abstractNote = {Multiphase flow is a critical process in a wide range of applications, including carbon sequestration, contaminant remediation, and groundwater management. Typically, this process is modeled by a nonlinear system of partial differential equations derived by considering the mass conservation of each phase (e.g., oil, water), along with constitutive laws for the relationship of phase velocity to phase pressure. In this study, we develop and study efficient solution algorithms for solving the algebraic systems of equations derived from a fully coupled and time-implicit treatment of models of multiphase flow. We explore the performance of several preconditioners based on algebraic multigrid (AMG) for solving the linearized problem, including “black-box” AMG applied directly to the system, a new version of constrained pressure residual multigrid (CPR-AMG) preconditioning, and a new preconditioner derived using an approximate Schur complement arising from the block factorization of the Jacobian. Finally, we show that the new methods are the most robust with respect to problem character, as determined by varying effects of capillary pressures, and we show that the block factorization preconditioner both is efficient and scales optimally with problem size.},

doi = {10.1137/16M1082652},

journal = {SIAM Journal on Scientific Computing},

number = [5],

volume = [39],

place = {United States},

year = {2017},

month = {10}

}

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