# Practical Applications

• ## Subsurface Flow Simulation

 What is subsurface flow simulation? Subsurface flow, in hydrology, is the flow of water beneath the earth’s surface that constitutes part of the water cycle. Numerical Results for Subsurface Flow Simulation Reference:  Cheng, Huang, Shu, X. Zhang, Zhang, and Zhou (2012)

• ## Reservoir Simulation

• What is Reservoir Simulation?

Reservoir simulation is the art of combining physics, mathematics, reservoir engineering, and computer programming to develop a tool for predicting hydrocarbon reservoir performance via various operating strategies. It is an important decision-making tool. For example, engineers use it to obtain information pertaining to the processes that take place in oil reservoirs. Such information enables an analysis of the various recovery strategies in order to effect optimal oil recovery. The crucial part of reservoir simulations is to solve large-scale discretized PDEs (highly coupled, nonsymmetric, and indefinite) over and over again. However, this is also the most time-consuming process of any modern petroleum reservoir simulator (more than 75%). The complexity of the geometry and of the physical model, heterogeneity, and size of reservoir model are continuing to grow, which makes these linear systems more difficult to solve using standard direct or iterative solvers.

A typical reservoir.

• The FASP Method for Reservoir Simulations

The FASP method takes full advantage of the underlying physical and analytic properties of the mathematical model.

 Transforms the complicated Jacobian system into three simpler auxiliary problems: an elliptic problem for the pressure variables, a hyperbolic problem for the saturation variables, and a purely algebraic problem for the well bottom-hole pressure variables. Thus, it can be used to design efficient and robust smoothers or preconditioners for each auxiliary problem. It then couples the auxiliary problems and applies the preconditioned Krylov subspace methods.

• Numerical Results for Reservoir Simulations

East Beverly Hill Field

Black oil model: 83,592 (129 $\times$ 72 $\times$ 9) cells, 994 faults, 169 wells.

JZ Oil Field

Polymer flooding: 474,297 cells, several faults, 37 wells.

SPE 10 Benchmark

Two phases (water and oil): 1.1 million cells, 5 wells.

Reference: X. Hu, W. Liu, G. Qin, J. Xu, Y. Yan, and C. Zhang (2011).

• ## Energy Storage

 Lithium Ion Battery Lithiumion batteries are rechargeable, and they are characterized by lithium ions that move from the negative electrode to the positive electrode during discharge and then back again during charging. Newton-Krylov-Multigrid Schemes for Lithium Ion Battery Simulation Finite volume method Newton’s method for the Butler-Volmer equation, and the whole nonlinear system Krylov subspace method (GMRes) with block Gauss-Seidel preconditioner Multigrid method for solving the Poisson-like problems in the preconditioner References: J. Wu, V. Srinivasan, X,  and C-Y, Wang (2002); J.Wu, X, and H. Zou (2006). Fuel Cells A fuel cell is a device that converts a fuel’s chemical energy from a fuel into electricity through a chemical reaction with oxygen or another oxidizing agent. Hydrogen is the most commonly used fuel for this purpose, but hydrocarbons such as natural gas and alcohols like methanol are sometimes used. Examples of fuel cells. Robust Methods for Fuel Cells Newton’s method for the nonlinear system Kirchhoff transformation Finite element-upwind finite volume method Nonoverlapping Schwarz domain decomposition method Overlapping domain decomposition method with non-matching grids Newton-Krylov-based solvers Numerical Results for Fuel Cells Fast convergence within 21 iterations versus oscillatory/nonconvergent iterations using commercial CFD software. References: P.Sun, G. Xue, C-Y. Wang, and Xu (2008);  P.Sun, G. Xue, C-Y. Wang, and Xu (2009); P.Sun, C-Y. Wang, and Xu (2010). Solar Energy Solar cells are devices that convert solar energy into electricity. Solar energy is a renewable source. Therefore, it is environmentally friendly. Example of solar cells.