Practical Applications

  • 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

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)X. Hu, J. Xu, and C. Zhang (2013)

  • Fluid Structure Interaction

  • What is FSI?

Fluid-structure interaction (FSI) aims at understanding the interaction between moving structure and fluid and how their interaction affects the interface between them. FSI has a wide range of applications in many areas including hemodynamics and wind/hydro turbines simulation. FSI problems are computationally challenging. The computational domain of FSI consists of fluid and structure subdomains. The position of the interface between fluid domain and structure domain is time dependent. Therefore, the shape of the fluid domain is one of the unknowns, increasing the nonlinearity of the FSI problems.

heart           Turek

Examples of fuel cells: (left) cardiac simulation; (right) 2D Turek benchmark

  • Monolithic Solver for FSI

Numerical solutions of FSI are roughly classified into partitioned approaches and monolithic approaches. Partitioned approaches employ single-physics solvers to solve the fluid and structure problems separately and then couple them by the interface conditions. Monolithic approaches solve the fluid and structure problems simultaneously. Monolithic approaches are considered more stable, although it is accompanied with larger linear systems and higher computational cost.
  • Lagrangian coordinate is used for structure equations, and Eulerian coordinate for fluid equations. We use ALE method to update the fluid mesh.
  • We discretize the coupled fluid and structure equations in a single linear system.
  • In order to solve it efficiently, we develop block preconditioners for this problem, which proves to be robust with respect to varying parameters and problems sizes.

  • Numerical Results for 2D Benchmark

Turek Fluid-Structure Interaction Benchmark


References: J. Xu and K. Yang (2014)

  • Hydroelectric Generator

Hydroelectric generator simulation involves the moving fluid domain, which is due to the rotation and deformation of the blade of the generator. Therefore, efficient monolithic solver for the couple system is difficult to construct. In order to simulation fluid coupled with rotating structures, like hydroturbine in hydroelectric generators, we develop a new ALE method to update the fluid mesh so that it can handle arbitrary rotation.

  • Numerical Results for Hydroelectric Generator

Moving mesh for real hydroelectric generator

  • Artificial Heart

Artificial heart is a kind of effective treatment for heart failure, which is the finial battlefield of cardiovascular disease. The artificial heart significantly changes the hemodynamics of the aorta. The main difficulty of artificial heart simulation is the complex interaction between blood, aorta and artificial heart.

Artificial heart in vessel

  • Numerical Results for Artificial Heart

Hemodynamics near the artificial heart

References: Q. Zhang, B. Gao, K. Gu, Y. Chang, J. Xu and P. Deuflhard (2014)

  • Magnetohydrodynamics

  • What is MHD?

The magnetohydrodynamics (MHD) model describes the dynamics of charged fluids in the presence of electromagnetic fields. A principle application of MHD is the modeling of plasma physics, ranging from plasma confinement for thermonuclear fusion to astrophysical plasma dynamics. MHD is also used to model the flow of liquid metals, for example, in magnetic pumps, liquid metal blankets in fusion reactor concepts, and aluminum electrolysis.


  • Structure-preserving Discretization and Robust Solver for MHD

  • We keep the electric field $E$ to form the mixed formulation of the MHD system, which preserves the divergence-free condition of the magnetic field exactly on the discrete level.
  • The block lower triangular preconditioner for FGMRes is applied.
  • When the time step size is small enough, robust number of iterations can be proved theoretically. And no such constrain required in actual tests.

  • Numerical Results for MHD

vstream            Tmag

Lid-driven cavity, Re=400, Rm=400: (left) stream line of the velocity; (right) distribution of the total magnetic field


Lid-driven cavity, Re=400, Rm=400: number of iterations

Reference: K. Hu, Y. Ma, and J. Xu (2015);  K. Hu, X. Hu, Y. Ma, and J. Xu (2015)

  • 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, J. Xu, 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.

fuelcell-1           fuelcell-2

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).

  • 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.

uw1            uw2

  • Numerical Results for Subsurface Flow Simulation


Reference:  J. Cheng, X. Huang, S. Shu, J. Xu, C. Zhang, S. Zhang, and Z. Zhou (2013)