# High-Enthalpy Geothermal Simulation with Continuous Localization in Physics

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methodology

## 3. Operator-Based Linearization (OBL) Approach

## 4. Single-Cell Problem with ‘Negative Compressibility’

#### 4.1. Formulations

- 1.
- Neglect the rock energy;
- 2.
- Heat conduction is ignored;
- 3.
- Rock is incompressible.

#### 4.2. Newton Path

#### 4.3. Operators

## 5. Continuous Localization in Physics

#### 5.1. Continuous Localization of the Newton Iteration

#### 5.2. Convergence Analysis

#### 5.3. One-Dimensional Test Case

#### 5.4. Two-Dimensional Test Case

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Newton path (dash line) starts from the initial condition; the contours (in ${l}_{2}$−norm) show the residuals; the blue dot represents the initial condition of the cell; the black dots are the Newton updates; the red dot shows the solution for a current timestep.

**Figure 2.**Newton paths (dash line) start from various points in pressure−enthalpy space; the contours (in ${l}_{2}$−norm) show the residuals, blue dots represent the initial guess, red dots show the solution for a current timestep.

**Figure 3.**Operators in mass and energy conservation equations for the single-cell problem. (

**a**) Mass accumulation operator, (

**b**) energy accumulation operator, (

**c**) mass flux operator, (

**d**) energy flux operator. The color changing from deep blue to red represents the magnitude of the operator values varying from low to high.

**Figure 4.**Newton path and residual contours (in ${l}_{2}$−norm) with continuation parameterization in physics: (

**a**) Newton path for a coarse resolution; (

**b**) Newton path for an intermediate resolution; (

**c**) Newton path for a fine (reference) resolution; the blue dot represents the initial guess, the black dot is the Newton update, the red dot shows the solution.

**Figure 6.**Newton path (dashed line) starting from various points in pressure-enthalpy space with (

**a**) moderate and (

**b**) large timestep; the blue dots represent the initial guess, the black dot is the solution of the coarsest resolution, the green dot is the intermediate solution and the red dot is the true solution; dashed lines in black, green and red represent the Newton path in the coarse, intermediate and fine (reference) physical resolutions, respectively; residual contours (in ${l}_{2}$−norm) are plotted for the reference physics.

**Figure 7.**Schematics of 1D test case. ${p}_{ini}$ and ${h}_{ini}$ refer to the initial pressure and enthalpy of the model, respectively, and ${s}_{g}$ refers to the initial steam saturation of the model. ${p}_{inj}$ and ${h}_{inj}$ are the injection pressure and enthalpy, and ${s}_{g\_inj}$ is the injection steam saturation.

**Figure 8.**Simulation results with continuous localization in physics with a large timestep (red line) and a conventional Newton-based approach with the reduced timestep (blue dots).

**Figure 9.**The porosity and permeability distribution of the two-dimensional setup. (

**a**) Porosity field; (

**b**) permeability field.

**Figure 10.**The pressure, enthalpy, temperature and water saturation maps of the model after one-year simulation with the proposed continuous localization of the Newton method.

**Figure 11.**The difference in the pressure, enthalpy, temperature and water saturation solutions between the conventional method and the proposed continuous localization of the Newton method with similar running timesteps.

Parameters | Conventional Method | Continuous Localization in Physics Method |
---|---|---|

Resolution of parameterization in (p, h) space | (128, 128) | (128, 4) » (128, 8) » (128, 32) » (128, 128) |

Total Newton iteration | 3325 | 683 |

Wasted Newton iteration | 760 | 80 |

CPU time, second | 14.9 | 4.2 |

Parameters | Unit | Values |
---|---|---|

Initial pressure | bar | 10 |

Initial enthalpy | kJ/kg | 1000 |

Initial water saturation | - | 0.04 |

Injection pressure | bar | 90 |

Injection enthalpy | kJ/kg | 100 |

Injection water saturation | - | 1.0 |

Parameters | Conventional Method | Continuous Localization in Physics Method |
---|---|---|

Resolution of parameterization in (p, h) space | (128, 128) | (128, 4) » (128, 8) » (128, 32) » (128, 128) |

Total Newton iteration | 2374 | 873 |

Wasted Newton iteration | 1140 | 280 |

CPU time, second | 96 | 37 |

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**MDPI and ACS Style**

Wang, Y.; Voskov, D.
High-Enthalpy Geothermal Simulation with Continuous Localization in Physics. *Mathematics* **2022**, *10*, 4328.
https://doi.org/10.3390/math10224328

**AMA Style**

Wang Y, Voskov D.
High-Enthalpy Geothermal Simulation with Continuous Localization in Physics. *Mathematics*. 2022; 10(22):4328.
https://doi.org/10.3390/math10224328

**Chicago/Turabian Style**

Wang, Yang, and Denis Voskov.
2022. "High-Enthalpy Geothermal Simulation with Continuous Localization in Physics" *Mathematics* 10, no. 22: 4328.
https://doi.org/10.3390/math10224328