@article{LEE2025126487,
title = {A phase-field diffraction model for thermo-hydro-mechanical propagating fractures},
journal = {International Journal of Heat and Mass Transfer},
volume = {239},
pages = {126487},
year = {2025},
issn = {0017-9310},
doi = {https://doi.org/10.1016/j.ijheatmasstransfer.2024.126487},
url = {https://www.sciencedirect.com/science/article/pii/S0017931024013152},
author = {Sanghyun Lee and Mary F. Wheeler and Thomas Wick},
keywords = {Thermo-poroelasticity, Phase-field fracture, Fixed-stress, Diffraction systems, Physics-based discretization, IPACS},
abstract = {This paper introduces a novel diffraction based thermo-hydraulic–mechanical (THM) model for fracture propagation using a phase-field fracture (PFF) approach. The key innovation of the THM-PFF model lies in its integrated treatment of four solution variables—displacements, phase-field, pressure, and temperature—each governed by a combination of conservation of momentum (mechanics problem), a variational inequality (constrained minimization problem), mass conservation (pressure problem), and energy conservation (temperature problem). This leads to a new formulation of a coupled variational inequality system. A major advancement is the development of an extended fixed-stress algorithm, where displacements, phase-field, pressures, and temperatures are solved in a staggered sequence. An important aspect of this work is the global coupling of pressures and temperatures across the domain using diffraction systems, with diffraction coefficients defined by material parameters weighted by the diffusive phase-field variable. To ensure robust local mass conservation, we employ enriched Galerkin finite elements (EG) for both pressure and temperature diffraction equations. By enriching the continuous Galerkin basis functions with discontinuous piecewise constants, EG accurately represents solution and parameter discontinuities while preserving local mass and energy conservation—crucial aspects for THM problems and realistic behavior. Moreover, the use of a predictor–corrector local mesh adaptivity scheme is employed, allowing the model to handle small phase-field length-scale parameters while maintaining high numerical accuracy and reasonable computational cost. These new model and algorithmic developments represent significant advances in the field and have been substantiated through rigorous numerical tests.}
}

This paper introduces a novel diffraction based thermo-hydraulic–mechanical (THM) model for fracture propagation using a phase-field fracture (PFF) approach. The key innovation of the THM-PFF model lies in its integrated treatment of four solution variables—displacements, phase-field, pressure, and temperature—each governed by a combination of conservation of momentum (mechanics problem), a variational inequality (constrained minimization problem), mass conservation (pressure problem), and energy conservation (temperature problem). This leads to a new formulation of a coupled variational inequality system. A major advancement is the development of an extended fixed-stress algorithm, where displacements, phase-field, pressures, and temperatures are solved in a staggered sequence. An important aspect of this work is the global coupling of pressures and temperatures across the domain using diffraction systems, with diffraction coefficients defined by material parameters weighted by the diffusive phase-field variable. To ensure robust local mass conservation, we employ enriched Galerkin finite elements (EG) for both pressure and temperature diffraction equations. By enriching the continuous Galerkin basis functions with discontinuous piecewise constants, EG accurately represents solution and parameter discontinuities while preserving local mass and energy conservation—crucial aspects for THM problems and realistic behavior. Moreover, the use of a predictor–corrector local mesh adaptivity scheme is employed, allowing the model to handle small phase-field length-scale parameters while maintaining high numerical accuracy and reasonable computational cost. These new model and algorithmic developments represent significant advances in the field and have been substantiated through rigorous numerical tests.

@article{JEONG2025303,
title = {Optimal control for Darcy's equation in a heterogeneous porous media},
journal = {Applied Numerical Mathematics},
volume = {207},
pages = {303-322},
year = {2025},
issn = {0168-9274},
doi = {https://doi.org/10.1016/j.apnum.2024.08.027},
url = {https://www.sciencedirect.com/science/article/pii/S0168927424002319},
author = {SeongHee Jeong and Sanghyun Lee},
keywords = {Optimal control, Darcy's flow, Heterogeneity,  interior penalty},
abstract = {In this paper, we investigate optimal control problems in heterogeneous porous media. The optimal control problem is governed by the Darcy's flow equation; where the pressure is the state variable and the source/sink is the control variable. Then we introduce the reduced optimal control problem which contains only the state variable by replacing the control variable with a dependent quantity of the state variable based on the Darcy's equation. Here we employ C0 interior penalty finite element methods for the spatial discretization to solve the reduced optimal control problem resulting in a fourth-order variational inequality. We use P2 Lagrange finite elements for C0 interior penalty methods, which require fewer degrees of freedom than C1 finite element methods. We provide a priori error estimates and stability analyses by considering a heterogeneous permeability coefficient. Several numerical examples validate the given theories and illustrate the capabilities of the proposed algorithm.}
}

In this paper, we investigate optimal control problems in heterogeneous porous media. The optimal control problem is governed by the Darcy's flow equation; where the pressure is the state variable and the source/sink is the control variable. Then we introduce the reduced optimal control problem which contains only the state variable by replacing the control variable with a dependent quantity of the state variable based on the Darcy's equation. Here we employ C0 interior penalty finite element methods for the spatial discretization to solve the reduced optimal control problem resulting in a fourth-order variational inequality. We use P2 Lagrange finite elements for C0 interior penalty methods, which require fewer degrees of freedom than C1 finite element methods. We provide a priori error estimates and stability analyses by considering a heterogeneous permeability coefficient. Several numerical examples validate the given theories and illustrate the capabilities of the proposed algorithm.

@article{LeeKuzminYang2024bound,
  title={Bound-preserving and entropy stable enriched Galerkin methods for nonlinear hyperbolic equations},
  author={Kuzmin, Dmitri and Lee, Sanghyun and Yang, Yi-Yung},
  journal={arXiv preprint arXiv:2411.19160},
  year={2024}
}

@article{lee2024thermo,
  title={A thermo-flow-mechanics-fracture model coupling a phase-field interface approach and thermo-fluid-structure interaction},
  author={Lee, Sanghyun and von Wahl, Henry and Wick, Thomas},
  journal={arXiv preprint arXiv:2409.03416},
  year={2024}
}

@article{dai2024thermal,
  title={A Thermal-Hydraulic-Mechanical-Chemical Coupling Model for Acid Fracture Propagation Based on a Phase-Field Method},
  author={Dai, Yifan and Hou, Bing and Lee, Sanghyun and Wick, Thomas},
  journal={Rock Mechanics and Rock Engineering},
  pages={1-23},
  year={2024},
  publisher={Springer}
}

@article{poudel2024novel,
  title={A novel technique for minimizing energy functional using neural networks},
  author={Poudel, Sanjeeb and Wang, Xiaoqiang and Lee, Sanghyun},
  journal={Engineering Applications of Artificial Intelligence},
  volume={133},
  pages={108313},
  year={2024},
  publisher={Elsevier}
}

@article{ballarin2024projection,
  title={Projection-based reduced order modeling of an iterative scheme for linear thermo-poroelasticity},
  author={Ballarin, Francesco and Lee, Sanghyun and Yi, Son-Young},
  journal={Results in Applied Mathematics},
  volume={21},
  pages={100430},
  year={2024},
  publisher={Elsevier}
}

@article{yi2024physics,
  title={Physics-preserving enriched Galerkin method for a fully-coupled thermo-poroelasticity model},
  author={Yi, Son-Young and Lee, Sanghyun},
  journal={Numerische Mathematik},
  pages={1--30},
  year={2024},
  publisher={Springer}
}

@article{LeeShin_SISC_2024,
author = {Lee, Sanghyun and Shin, Yeonjong},
title = {On the Training and Generalization of Deep Operator Networks},
journal = {SIAM Journal on Scientific Computing},
volume = {46},
number = {4},
pages = {C273-C296},
year = {2024},
doi = {10.1137/23M1598751},
URL = {https://doi.org/10.1137/23M1598751},
eprint = {https://doi.org/10.1137/23M1598751},
abstract = { Abstract. We present a novel training method for deep operator networks (DeepONets), one of the most popular neural network models for operators. DeepONets are constructed by two subnetworks, namely the branch and trunk networks. Typically, the two subnetworks are trained simultaneously, which amounts to solving a complex optimization problem in a high dimensional space. In addition, the nonconvex and nonlinear nature makes training very challenging. To tackle such a challenge, we propose a two-step training method that trains the trunk network first and then sequentially trains the branch network. The core mechanism is motivated by the divide-and-conquer paradigm and is the decomposition of the entire complex training task into two subtasks with reduced complexity. Therein the Gram–Schmidt orthonormalization process is introduced which significantly improves stability and generalization ability. On the theoretical side, we establish a generalization error estimate in terms of the number of training data, the width of DeepONets, and the number of input and output sensors. Numerical examples are presented to demonstrate the effectiveness of the two-step training method, including Darcy flow in heterogeneous porous media. }
}

Abstract. We present a novel training method for deep operator networks (DeepONets), one of the most popular neural network models for operators. DeepONets are constructed by two subnetworks, namely the branch and trunk networks. Typically, the two subnetworks are trained simultaneously, which amounts to solving a complex optimization problem in a high dimensional space. In addition, the nonconvex and nonlinear nature makes training very challenging. To tackle such a challenge, we propose a two-step training method that trains the trunk network first and then sequentially trains the branch network. The core mechanism is motivated by the divide-and-conquer paradigm and is the decomposition of the entire complex training task into two subtasks with reduced complexity. Therein the Gram–Schmidt orthonormalization process is introduced which significantly improves stability and generalization ability. On the theoretical side, we establish a generalization error estimate in terms of the number of training data, the width of DeepONets, and the number of input and output sensors. Numerical examples are presented to demonstrate the effectiveness of the two-step training method, including Darcy flow in heterogeneous porous media.

@article{LeeKadHam_2024,
author = {Lee , Sanghyun and Kadeethum , Teeratorn and M. Nick , Hamidreza},
title = {Choice of Interior Penalty Coefficient for Interior Penalty Discontinuous Galerkin Method for Biot’s System by Employing Machine Learning},
journal = {International Journal of Numerical Analysis and Modeling},
year = {2024},
volume = {21},
number = {5},
pages = {764--792},
abstract = {
This paper uses neural networks and machine learning to study the optimal choice of
the interior penalty parameter of the discontinuous Galerkin finite element methods for both the
elliptic problems and Biot’s systems. It is crucial to choose the optimal interior penalty parameter,
which is not too small or too large for the stability, robustness, and efficiency of the approximated numerical solutions. Both linear regression and nonlinear artificial neural network methods
are employed and compared using several numerical experiments to illustrate the capability of
our proposed computational framework. This framework is integral to developing automated numerical simulation because it can automatically identify the optimal interior penalty parameter.
Real-time feedback could also be implemented to update and improve model accuracy on the fly.
},
issn = {2617-8710},
doi = {https://doi.org/10.4208/ijnam2024-1031},
url = {http://global-sci.org/intro/article_detail/ijnam/23452.html}
}

This paper uses neural networks and machine learning to study the optimal choice of the interior penalty parameter of the discontinuous Galerkin finite element methods for both the elliptic problems and Biot’s systems. It is crucial to choose the optimal interior penalty parameter, which is not too small or too large for the stability, robustness, and efficiency of the approximated numerical solutions. Both linear regression and nonlinear artificial neural network methods are employed and compared using several numerical experiments to illustrate the capability of our proposed computational framework. This framework is integral to developing automated numerical simulation because it can automatically identify the optimal interior penalty parameter. Real-time feedback could also be implemented to update and improve model accuracy on the fly.

@article{lee2023locking,
  title={Locking-Free and Locally-Conservative Enriched Galerkin Method for Poroelasticity},
  author={Lee, Sanghyun and Yi, Son-Young},
  journal={Journal of Scientific Computing},
  volume={94},
  number={1},
  pages={1--23},
  year={2023},
  publisher={Springer}
}

@article{kadeethum2022enhancing,
  title={Enhancing high-fidelity nonlinear solver with reduced order model},
  author={Kadeethum, Teeratorn and O’malley, Daniel and Ballarin, Francesco and Ang, Ida and Fuhg, Jan N and Bouklas, Nikolaos and Silva, Vinicius LS and Salinas, Pablo and Heaney, Claire E and Pain, Christopher C and others},
  journal={Scientific Reports},
  volume={12},
  number={1},
  pages={1-15},
  year={2022},
  publisher={Nature Publishing Group}
}

@article{LEE2022113715,
title = {Finite element simulation of quasi-static tensile fracture in nonlinear strain-limiting solids with the phase-field approach},
journal = {Journal of Computational and Applied Mathematics},
volume = {399},
pages = {113715},
year = {2022},
issn = {0377-0427},
doi = {https://doi.org/10.1016/j.cam.2021.113715},
url = {https://www.sciencedirect.com/science/article/pii/S037704272100337X},
author = {Sanghyun Lee and Hyun Chul Yoon and S.M. Mallikarjunaiah},
keywords = {Strain-limiting model, Nonlinear elasticity, LEFM, Fracture propagation, Phase-field, Finite element method},
abstract = {We investigate a quasi-static tensile fracture in nonlinear strain-limiting solids by coupling with the phase-field approach. A classical model for the growth of fractures in an elastic material is formulated in the framework of linear elasticity for deformation systems. This linear elastic fracture mechanics (LEFM) model is derived based on the assumption of small strain. However, the boundary value problem formulated within the LEFM and under traction-free boundary conditions predicts large singular crack-tip strains. Fundamentally, this result is directly in contradiction with the underlying assumption of small strain. In this work, we study a theoretical framework of nonlinear strain-limiting models, which are algebraic nonlinear relations between stress and strain. These models are consistent with the basic assumption of small strain. The advantage of such framework over the LEFM is that the strain remains bounded even if the crack-tip stress tends to the infinity. Then, employing the phase-field approach, the distinct predictions for tensile crack growth can be governed by the model. Several numerical examples to evaluate the efficacy and the performance of the model and numerical algorithms structured on finite element method are presented. Detailed comparisons of the strain, fracture energy with corresponding discrete propagation speed between the nonlinear strain-limiting model and the LEFM for the quasi-static tensile fracture are discussed.}
}

We investigate a quasi-static tensile fracture in nonlinear strain-limiting solids by coupling with the phase-field approach. A classical model for the growth of fractures in an elastic material is formulated in the framework of linear elasticity for deformation systems. This linear elastic fracture mechanics (LEFM) model is derived based on the assumption of small strain. However, the boundary value problem formulated within the LEFM and under traction-free boundary conditions predicts large singular crack-tip strains. Fundamentally, this result is directly in contradiction with the underlying assumption of small strain. In this work, we study a theoretical framework of nonlinear strain-limiting models, which are algebraic nonlinear relations between stress and strain. These models are consistent with the basic assumption of small strain. The advantage of such framework over the LEFM is that the strain remains bounded even if the crack-tip stress tends to the infinity. Then, employing the phase-field approach, the distinct predictions for tensile crack growth can be governed by the model. Several numerical examples to evaluate the efficacy and the performance of the model and numerical algorithms structured on finite element method are presented. Detailed comparisons of the strain, fracture energy with corresponding discrete propagation speed between the nonlinear strain-limiting model and the LEFM for the quasi-static tensile fracture are discussed.

@article{bing2022numerical,
  title={Numerical simulation of pores connection by acid fracturing based on phase field method},
  author={Bing, Hou and Dai Yifan, Fan Meng and Kunpeng, Zhang and Thomas, Wick and Sanghyun, Lee},
  journal={Acta Petrolei Sinica},
  volume={43},
  number={6},
  pages={849},
  year={2022}
}

@article{YI2022115,
title = {An enriched Galerkin method for the Stokes equations},
journal = {Computers & Mathematics with Applications},
volume = {120},
pages = {115-131},
year = {2022},
issn = {0898-1221},
doi = {https://doi.org/10.1016/j.camwa.2022.06.018},
url = {https://www.sciencedirect.com/science/article/pii/S0898122122002632},
author = {Son-Young Yi and Xiaozhe Hu and Sanghyun Lee and James H. Adler},
keywords = {Enriched Galerkin, Stokes, Finite element method, Inf-sup},
abstract = {We present a new enriched Galerkin (EG) scheme for the Stokes equations based on piecewise linear elements for the velocity unknowns and piecewise constant elements for the pressure. The proposed EG method augments the conforming piecewise linear space for velocity by adding an additional degree of freedom which corresponds to one discontinuous linear basis function per element. Thus, the total number of degrees of freedom is significantly reduced in comparison with standard conforming, non-conforming, and discontinuous Galerkin schemes for the Stokes equation. We show the well-posedness of the new EG approach and prove that the scheme converges optimally. For the solution of the resulting large-scale indefinite linear systems we propose robust block preconditioners, yielding scalable results independent of the discretization and physical parameters. Numerical results confirm the convergence rates of the discretization and also the robustness of the linear solvers for a variety of test problems.}
}

We present a new enriched Galerkin (EG) scheme for the Stokes equations based on piecewise linear elements for the velocity unknowns and piecewise constant elements for the pressure. The proposed EG method augments the conforming piecewise linear space for velocity by adding an additional degree of freedom which corresponds to one discontinuous linear basis function per element. Thus, the total number of degrees of freedom is significantly reduced in comparison with standard conforming, non-conforming, and discontinuous Galerkin schemes for the Stokes equation. We show the well-posedness of the new EG approach and prove that the scheme converges optimally. For the solution of the resulting large-scale indefinite linear systems we propose robust block preconditioners, yielding scalable results independent of the discretization and physical parameters. Numerical results confirm the convergence rates of the discretization and also the robustness of the linear solvers for a variety of test problems.

@article{yi2022locking,
  title={Locking-free enriched Galerkin method for linear elasticity},
  author={Yi, Son-Young and Lee, Sanghyun and Zikatanov, Ludmil},
  journal={SIAM Journal on Numerical Analysis},
  volume={60},
  number={1},
  pages={52-75},
  year={2022},
  publisher={SIAM},
abstract = { We propose a new locking-free enriched Galerkin method for solving the linear elasticity problem. The method is based on the discontinuous Galerkin formulation, but its approximation space is a continuous piecewise linear vector-valued function space enriched by some discontinuous piecewise linear functions. An a priori error estimate of optimal order in the energy norm is proved and shown to be independent of a Lamé parameter \lambda, hence the proposed method is free of volumetric locking when modeling incompressible materials. Moreover, a uniform preconditioner with respect to the mesh size is established in the operator preconditioning framework. We provide several numerical examples to confirm the accuracy and the robustness of the new method and demonstrate a good performance of the preconditioner. }
}

We propose a new locking-free enriched Galerkin method for solving the linear elasticity problem. The method is based on the discontinuous Galerkin formulation, but its approximation space is a continuous piecewise linear vector-valued function space enriched by some discontinuous piecewise linear functions. An a priori error estimate of optimal order in the energy norm is proved and shown to be independent of a Lamé parameter λ, hence the proposed method is free of volumetric locking when modeling incompressible materials. Moreover, a uniform preconditioner with respect to the mesh size is established in the operator preconditioning framework. We provide several numerical examples to confirm the accuracy and the robustness of the new method and demonstrate a good performance of the preconditioner.

@article{KNLB_2021_JCP,
title = "Enriched Galerkin discretization for modeling poroelasticity and permeability alteration in heterogeneous porous media",
journal = "Journal of Computational Physics",
volume = "427",
pages = "110030",
year = "2021",
issn = "0021-9991",
doi = "https://doi.org/10.1016/j.jcp.2020.110030",
url = "http://www.sciencedirect.com/science/article/pii/S0021999120308044",
author = "T. Kadeethum and H.M. Nick and S. Lee and F. Ballarin",
keywords = "Deformable porous media, Poroelastic effects, Biot's system, Enriched Galerkin, Finite element method, Heterogeneity",
abstract = "In this paper, we utilize the enriched Galerkin (EG) finite element method for the flow equation in Biot's system, which provides a robust locally conservative flux in heterogeneous porous media. The computational algorithm to solve the coupled system with the permeability alteration is presented with the linearization and Picard's iterative scheme. The block structure is utilized for the linear system in numerical discretization, and the computer code is shared in the open-source platform. In the numerical experiments, we compare the proposed EG method with the classical continuous Galerkin (CG) and discontinuous Galerkin (DG) finite element methods in different scenarios, including the North sea reservoirs setup. While DG and EG methods provide similar approximations for the pressure solutions, the CG method produces spurious oscillations in fluid pressure and volumetric strain solutions near the material interfaces, especially for the soft materials. The difference of flux approximation between EG and DG methods is insignificant; still, the EG method demands approximately two and three times fewer degrees of freedom than the DG method for two- and three-dimensional geometries."
}

In this paper, we utilize the enriched Galerkin (EG) finite element method for the flow equation in Biot's system, which provides a robust locally conservative flux in heterogeneous porous media. The computational algorithm to solve the coupled system with the permeability alteration is presented with the linearization and Picard's iterative scheme. The block structure is utilized for the linear system in numerical discretization, and the computer code is shared in the open-source platform. In the numerical experiments, we compare the proposed EG method with the classical continuous Galerkin (CG) and discontinuous Galerkin (DG) finite element methods in different scenarios, including the North sea reservoirs setup. While DG and EG methods provide similar approximations for the pressure solutions, the CG method produces spurious oscillations in fluid pressure and volumetric strain solutions near the material interfaces, especially for the soft materials. The difference of flux approximation between EG and DG methods is insignificant; still, the EG method demands approximately two and three times fewer degrees of freedom than the DG method for two- and three-dimensional geometries.

@inproceedings{lee2021physics,
  title={Physics-informed Neural Networks for Solving Coupled Flow and Transport System.},
  author={Lee, Sanghyun and Kadeethum, Teeratorn},
  booktitle={AAAI Spring Symposium: MLPS},
  year={2021}
}

@article{KADEETHUM2021104774,
title = {A locally conservative mixed finite element framework for coupled hydro-mechanical-chemical processes in heterogeneous porous media},
journal = {Computers \& Geosciences},
volume = {152},
pages = {104774},
year = {2021},
issn = {0098-3004},
doi = {https://doi.org/10.1016/j.cageo.2021.104774},
url = {https://www.sciencedirect.com/science/article/pii/S0098300421000790},
author = {T. Kadeethum and S. Lee and F. Ballarin and J. Choo and H.M. Nick},
keywords = {Hydro-mechanical-chemical coupling, Poroelasticity, Reactive flow, Mixed finite element method, Enriched Galerkin method, Local conservation},
abstract = {This paper presents a mixed finite element framework for coupled hydro-mechanical–chemical processes in heterogeneous porous media. The framework combines two types of locally conservative discretization schemes: (1) an enriched Galerkin method for reactive flow, and (2) a three-field mixed finite element method for coupled fluid flow and solid deformation. This combination ensures local mass conservation, which is critical to flow and transport in heterogeneous porous media, with a relatively affordable computational cost. A particular class of the framework is constructed for calcite precipitation/dissolution reactions, incorporating their nonlinear effects on the fluid viscosity and solid deformation. Linearization schemes and algorithms for solving the nonlinear algebraic system are also presented. Through numerical examples of various complexity, we demonstrate that the proposed framework is a robust and efficient computational method for simulation of reactive flow and transport in deformable porous media, even when the material properties are strongly heterogeneous and anisotropic.}
}

This paper presents a mixed finite element framework for coupled hydro-mechanical–chemical processes in heterogeneous porous media. The framework combines two types of locally conservative discretization schemes: (1) an enriched Galerkin method for reactive flow, and (2) a three-field mixed finite element method for coupled fluid flow and solid deformation. This combination ensures local mass conservation, which is critical to flow and transport in heterogeneous porous media, with a relatively affordable computational cost. A particular class of the framework is constructed for calcite precipitation/dissolution reactions, incorporating their nonlinear effects on the fluid viscosity and solid deformation. Linearization schemes and algorithms for solving the nonlinear algebraic system are also presented. Through numerical examples of various complexity, we demonstrate that the proposed framework is a robust and efficient computational method for simulation of reactive flow and transport in deformable porous media, even when the material properties are strongly heterogeneous and anisotropic.

@misc{lee2020choice,
      title={Choice of Interior Penalty Coefficient for Interior Penalty Discontinuous Galerkin Method for Biot's System by Employing Machine Learning},
      author={Sanghyun Lee and Teeratorn Kadeethum and Hamidreza M. Nick},
      year={2021},
      archivePrefix={arXiv},
      primaryClass={math.NA}
}

@article{KNLB_2021_JCP,
title = "Enriched Galerkin discretization for modeling poroelasticity and permeability alteration in heterogeneous porous media",
journal = "Journal of Computational Physics",
volume = "427",
pages = "110030",
year = "2021",
issn = "0021-9991",
doi = "https://doi.org/10.1016/j.jcp.2020.110030",
url = "http://www.sciencedirect.com/science/article/pii/S0021999120308044",
author = "T. Kadeethum and H.M. Nick and S. Lee and F. Ballarin",
keywords = "Deformable porous media, Poroelastic effects, Biot's system, Enriched Galerkin, Finite element method, Heterogeneity",
abstract = "In this paper, we utilize the enriched Galerkin (EG) finite element method for the flow equation in Biot's system, which provides a robust locally conservative flux in heterogeneous porous media. The computational algorithm to solve the coupled system with the permeability alteration is presented with the linearization and Picard's iterative scheme. The block structure is utilized for the linear system in numerical discretization, and the computer code is shared in the open-source platform. In the numerical experiments, we compare the proposed EG method with the classical continuous Galerkin (CG) and discontinuous Galerkin (DG) finite element methods in different scenarios, including the North sea reservoirs setup. While DG and EG methods provide similar approximations for the pressure solutions, the CG method produces spurious oscillations in fluid pressure and volumetric strain solutions near the material interfaces, especially for the soft materials. The difference of flux approximation between EG and DG methods is insignificant; still, the EG method demands approximately two and three times fewer degrees of freedom than the DG method for two- and three-dimensional geometries."
}

In this paper, we utilize the enriched Galerkin (EG) finite element method for the flow equation in Biot's system, which provides a robust locally conservative flux in heterogeneous porous media. The computational algorithm to solve the coupled system with the permeability alteration is presented with the linearization and Picard's iterative scheme. The block structure is utilized for the linear system in numerical discretization, and the computer code is shared in the open-source platform. In the numerical experiments, we compare the proposed EG method with the classical continuous Galerkin (CG) and discontinuous Galerkin (DG) finite element methods in different scenarios, including the North sea reservoirs setup. While DG and EG methods provide similar approximations for the pressure solutions, the CG method produces spurious oscillations in fluid pressure and volumetric strain solutions near the material interfaces, especially for the soft materials. The difference of flux approximation between EG and DG methods is insignificant; still, the EG method demands approximately two and three times fewer degrees of freedom than the DG method for two- and three-dimensional geometries.

@article{YoonLeeMudd2021,
author={Yoon, Hyun C.
and Lee, Sanghyun
and Mallikarjunaiah, S. M.},
title={Quasi-static anti-plane shear crack propagation in nonlinear strain-limiting elastic solids using phase-field approach},
journal={International Journal of Fracture},
year={2021},
month={Jan},
day={02},
abstract={We study a quasi-static evolution of anti-plane crack with the nonlinear strain-limiting model using the phase-field approach. The nonlinear strain-limiting models, a subclass of the implicit constitutive relations, allow the linearized strain value to remain small even if the stress value tends to infinity. To compute the quasi-static crack, we solve the constrained energy minimization for the nonlinear bulk energy coupled with diffusive crack energy. An iterative staggered scheme (L-scheme) is employed for coupling the nonlinear mechanics and phase-field, and augmented Lagrangian is utilized to accommodate crack irreversibility. Several numerical experiments of the proposed framework substantiate the bounded strain in the neighborhood of the crack-tip for both static and quasi-static cracks. The comparisons of bulk and the crack energies and the discrete crack-tip speed between the LEFM and our model are presented.},
issn={1573-2673},
doi={10.1007/s10704-020-00501-y},
url={https://doi.org/10.1007/s10704-020-00501-y}
}

We study a quasi-static evolution of anti-plane crack with the nonlinear strain-limiting model using the phase-field approach. The nonlinear strain-limiting models, a subclass of the implicit constitutive relations, allow the linearized strain value to remain small even if the stress value tends to infinity. To compute the quasi-static crack, we solve the constrained energy minimization for the nonlinear bulk energy coupled with diffusive crack energy. An iterative staggered scheme (L-scheme) is employed for coupling the nonlinear mechanics and phase-field, and augmented Lagrangian is utilized to accommodate crack irreversibility. Several numerical experiments of the proposed framework substantiate the bounded strain in the neighborhood of the crack-tip for both static and quasi-static cracks. The comparisons of bulk and the crack energies and the discrete crack-tip speed between the LEFM and our model are presented.

@article{lee2020choice,
  title={Choice of Interior Penalty Coefficient for Interior Penalty Discontinuous Galerkin Method for Biot's System by Employing Machine Learning},
  author={Lee, Sanghyun and Kadeethum, Teeratorn and Nick, Hamidreza M},
  journal={arXiv preprint arXiv:2007.10119},
  year={2020}
}

@article{Wheeler_Lee_2020,
author = {Wheeler, Mary F. and Lee, Sanghyun},
title = {Hydraulic Fracture Propagation Simulations in Porous Media with Natural Fractures by IPACS},
journal = {Unconventional Resources Technology Conference, Virtual, 20-22 July 2020},
chapter = {},
pages = {793-798},
year = {2020},
doi = {10.15530/urtec-2020-2927},
URL = {https://library.seg.org/doi/abs/10.15530/urtec-2020-2927},
eprint = {https://library.seg.org/doi/pdf/10.15530/urtec-2020-2927},
    abstract = { Recent studies reveal that unconventional reservoirs contain complex natural fracture networks. Thus, in stimulating hydraulic fractures, it is important to study the interactions between natural fractures and hydraulic fractures. The goal of this work is to describe practical aspects of recent advances in the domain of geomechanical and discrete fracture network coupling, stimulation modeling, treating large number of fractures, adaptive mesh refinement methods, and overall fracture management within an unconventional setting. The computational framework is developed as an in-house code named IPACS (Integrated Phase-field Advanced Crack Propagation Simulator). Here, we describe a diffusive adaptive finite element phase field approach for modeling natural and hydraulic fractures. High fidelity finite element methods are employed to model multiphase flow with local mass conservation and dynamic mesh adaptivity. Geomechanics approximated by a continuous Galerkin finite element method is coupled to multiphase flow approximated by an enriched Galerkin finite element method by applying an iteratively coupled scheme. }
}

Recent studies reveal that unconventional reservoirs contain complex natural fracture networks. Thus, in stimulating hydraulic fractures, it is important to study the interactions between natural fractures and hydraulic fractures. The goal of this work is to describe practical aspects of recent advances in the domain of geomechanical and discrete fracture network coupling, stimulation modeling, treating large number of fractures, adaptive mesh refinement methods, and overall fracture management within an unconventional setting. The computational framework is developed as an in-house code named IPACS (Integrated Phase-field Advanced Crack Propagation Simulator). Here, we describe a diffusive adaptive finite element phase field approach for modeling natural and hydraulic fractures. High fidelity finite element methods are employed to model multiphase flow with local mass conservation and dynamic mesh adaptivity. Geomechanics approximated by a continuous Galerkin finite element method is coupled to multiphase flow approximated by an enriched Galerkin finite element method by applying an iteratively coupled scheme.

@article{kadeethum2020finite,
  title={Finite element solvers for biot's poroelasticity equations in porous media},
  author={Kadeethum, Teeratorn and Lee, S and Nick, HM},
  journal={Mathematical Geosciences},
  volume={52},
  number={8},
  pages={977-1015},
  year={2020},
  publisher={Springer}
}

@article{RuppLee_EG_2020,
 author={A. Rupp, S. Lee},
 journal ="Journal of Scientific Computing", 
 volume ="84",
 pages ="9",
 title={Continuous Galerkin and enriched Galerkin methods with arbitrary order discontinuous trial functions for the elliptic and parabolic problems with jump conditions},
 year={2020}
}

@article{KadNickLeeBallarin_2019_mixed,
title = "Flow in Porous Media with Low Dimensional Fractures by Employing Enriched Galerkin Method",
journal = "Advances in Water Resources",
pages = "103620",
year = "2020",
issn = "0309-1708",
doi = "https://doi.org/10.1016/j.advwatres.2020.103620",
url = "http://www.sciencedirect.com/science/article/pii/S0309170819312576",
author = "Kadeethum, T. and Nick, H.M. and Lee, S. and Ballarin, F.",
keywords = "Fractured porous media, Mixed-dimensional, Enriched Galerkin, Finite element method, Heterogeneity, Local mass conservative",
abstract = "This paper presents the enriched Galerkin discretization for modeling fluid flow in fractured porous media using the mixed-dimensional approach. The proposed method has been tested against published benchmarks. Since fracture and porous media discontinuities can significantly influence single- and multi-phase fluid flow, the heterogeneous and anisotropic matrix permeability setting is utilized to assess the enriched Galerkin performance in handling the discontinuity within the matrix domain and between the matrix and fracture domains. Our results illustrate that the enriched Galerkin method has the same advantages as the discontinuous Galerkin method; for example, it conserves local and global fluid mass, captures the pressure discontinuity, and provides the optimal error convergence rate. However, the enriched Galerkin method requires much fewer degrees of freedom than the discontinuous Galerkin method in its classical form. The pressure solutions produced by both methods are similar regardless of the conductive or non-conductive fractures or heterogeneity in matrix permeability. This analysis shows that the enriched Galerkin scheme reduces the computational costs while offering the same accuracy as the discontinuous Galerkin so that it can be applied for large-scale flow problems. Furthermore, the results of a time-dependent problem for a three-dimensional geometry reveal the value of correctly capturing the discontinuities as barriers or highly-conductive fractures."
}

This paper presents the enriched Galerkin discretization for modeling fluid flow in fractured porous media using the mixed-dimensional approach. The proposed method has been tested against published benchmarks. Since fracture and porous media discontinuities can significantly influence single- and multi-phase fluid flow, the heterogeneous and anisotropic matrix permeability setting is utilized to assess the enriched Galerkin performance in handling the discontinuity within the matrix domain and between the matrix and fracture domains. Our results illustrate that the enriched Galerkin method has the same advantages as the discontinuous Galerkin method; for example, it conserves local and global fluid mass, captures the pressure discontinuity, and provides the optimal error convergence rate. However, the enriched Galerkin method requires much fewer degrees of freedom than the discontinuous Galerkin method in its classical form. The pressure solutions produced by both methods are similar regardless of the conductive or non-conductive fractures or heterogeneity in matrix permeability. This analysis shows that the enriched Galerkin scheme reduces the computational costs while offering the same accuracy as the discontinuous Galerkin so that it can be applied for large-scale flow problems. Furthermore, the results of a time-dependent problem for a three-dimensional geometry reveal the value of correctly capturing the discontinuities as barriers or highly-conductive fractures.

@article{IPACS_2020,
title = "IPACS: Integrated Phase-Field Advanced Crack Propagation Simulator. An adaptive, parallel, physics-based-discretization phase-field framework for fracture propagation in porous media",
journal = "Computer Methods in Applied Mechanics and Engineering",
volume = "367",
pages = "113124",
year = "2020",
issn = "0045-7825",
doi = "https://doi.org/10.1016/j.cma.2020.113124",
url = "http://www.sciencedirect.com/science/article/pii/S0045782520303091",
author = "Mary F. Wheeler and Thomas Wick and Sanghyun Lee",
keywords = "Phase-field fracture, Porous media, Computer implementation, Numerical simulations, Handbook, IPACS",
abstract = "In this work, we review and describe our computational framework for solving multiphysics phase-field fracture problems in porous media. Therein, the following five coupled nonlinear physical models are addressed: displacements (geo-mechanics), a phase-field variable to indicate the fracture position, a pressure equation (to describe flow), a proppant concentration equation, and/or a saturation equation for two-phase fracture flow, and finally a finite element crack width problem. The overall coupled problem is solved with a staggered solution approach, known in subsurface modeling as the fixed-stress iteration. A main focus is on physics-based discretizations. Galerkin finite elements are employed for the displacement-phase-field system and the crack width problem. Enriched Galerkin formulations are used for the pressure equation. Further enrichments using entropy-vanishing viscosity are employed for the proppant and/or saturation equations. A robust and efficient quasi-monolithic semi-smooth Newton solver, local mesh adaptivity, and parallel implementations allow for competitive timings in terms of the computational cost. Our framework can treat two- and three-dimensional realistic field and laboratory examples. The resulting program is an in-house code named IPACS (Integrated Phase-field Advanced Crack Propagation Simulator) and is based on the finite element library deal.II. Representative numerical examples are included in this document."
}

In this work, we review and describe our computational framework for solving multiphysics phase-field fracture problems in porous media. Therein, the following five coupled nonlinear physical models are addressed: displacements (geo-mechanics), a phase-field variable to indicate the fracture position, a pressure equation (to describe flow), a proppant concentration equation, and/or a saturation equation for two-phase fracture flow, and finally a finite element crack width problem. The overall coupled problem is solved with a staggered solution approach, known in subsurface modeling as the fixed-stress iteration. A main focus is on physics-based discretizations. Galerkin finite elements are employed for the displacement-phase-field system and the crack width problem. Enriched Galerkin formulations are used for the pressure equation. Further enrichments using entropy-vanishing viscosity are employed for the proppant and/or saturation equations. A robust and efficient quasi-monolithic semi-smooth Newton solver, local mesh adaptivity, and parallel implementations allow for competitive timings in terms of the computational cost. Our framework can treat two- and three-dimensional realistic field and laboratory examples. The resulting program is an in-house code named IPACS (Integrated Phase-field Advanced Crack Propagation Simulator) and is based on the finite element library deal.II. Representative numerical examples are included in this document.

@article{ChoiLee_2019,
title = "Optimal error estimate of elliptic problems with Dirac sources for discontinuous and enriched Galerkin methods",
journal = "Applied Numerical Mathematics",
volume = "150",
pages = "76 - 104",
year = "2020",
issn = "0168-9274",
doi = "https://doi.org/10.1016/j.apnum.2019.09.010",
url = "http://www.sciencedirect.com/science/article/pii/S0168927419302491",
author = "Woocheol Choi and Sanghyun Lee",
keywords = "Singularity, Dirac source, Discontinuous Galerkin finite element methods, Enriched Galerkin finite element methods, A priori estimates",
abstract = "We present an optimal a priori error estimates of the elliptic problems with Dirac sources away from the singular point using discontinuous and enriched Galerkin finite element methods. It is widely shown that the finite element solutions for elliptic problems with Dirac source terms converge sub-optimally in classical norms on uniform meshes. However, here we employ inductive estimates and L2 norm to obtain the optimal order by excluding the small ball regions with the singularities for both two and three dimensional domains. Numerical examples are presented to substantiate our theoretical results."
}

We present an optimal a priori error estimates of the elliptic problems with Dirac sources away from the singular point using discontinuous and enriched Galerkin finite element methods. It is widely shown that the finite element solutions for elliptic problems with Dirac source terms converge sub-optimally in classical norms on uniform meshes. However, here we employ inductive estimates and L2 norm to obtain the optimal order by excluding the small ball regions with the singularities for both two and three dimensional domains. Numerical examples are presented to substantiate our theoretical results.

@article{LeeWheeler_2020,
  title={Modeling interactions of natural and two-phase fluid-filled fracture propagation in porous media},
  author={Lee, Sanghyun and Wheeler, Mary F},
  journal={Computational Geosciences},
  pages={1-25},
  year={2020},
  publisher={Springer}
}

@inproceedings{KadNickLee_2019_ARMA,
  title={A Novel Enriched Galerkin Method for Modelling Coupled Flow and Mechanical Deformation in Heterogeneous Porous Media},
  author={T. Kadeethum, H.M. Nick, S. Lee, C.N. Richardson, S. Salimzadeh, F. Ballarin},
  booktitle={American Rock Mechanics Association},
  year={2019},
  organization={American Rock Mechanics Association}
}

@inproceedings{KadNickLee_2019_IAMG,
  title={Comparison of Two- and Three-field Formulation Discretizations for Flow and Solid Deformation in Heterogeneous Porous Media},
  author ={T. Kadeethum, H. M. Nick, and S. Lee},
  booktitle={International Association for Mathematical Geosciences},
  year={2019},
  organization={International Association for Mathematical Geosciences}
}

@inproceedings{dong2019numerical,
  title={Numerical Simulation of Matrix Acidizing in Fractured Carbonate Reservoirs Using Adaptive Enriched Galerkin Method},
  author={Dong, Rencheng and Lee, Sanghyun and Wheeler, Mary and others},
  booktitle={SPE Reservoir Simulation Conference},
  year={2019},
  organization={Society of Petroleum Engineers}
}

@inproceedings{wheeler2019unconventional,
  title={Unconventional Reservoir Management Modeling Coupling Diffusive Zone/Phase Field Fracture Modeling and Fracture Probability Maps},
  author={Wheeler, Mary F and Srinivasan, Sanjay and Lee, Sanghyun and Singh, Manik and others},
  booktitle={SPE Reservoir Simulation Conference},
  year={2019},
  organization={Society of Petroleum Engineers}
}

@article{SogoLeeWheeler_2019,
author = {Shiozawa, Sogo and Lee, Sanghyun and Wheeler, Mary F.},
title = {The effect of stress boundary conditions on fluid-driven fracture propagation in porous media using a phase-field modeling approach},
journal = {International Journal for Numerical and Analytical Methods in Geomechanics},
year = {2019},
volume = {43},
number = {6},
pages = {1316-1340},
keywords = {fluid-driven fracture, phase field, porous media, stress boundary conditions},
doi = {10.1002/nag.2899},
url = {https://onlinelibrary.wiley.com/doi/abs/10.1002/nag.2899},
eprint = {https://onlinelibrary.wiley.com/doi/pdf/10.1002/nag.2899},
abstract = {Summary A phase-field approach for fluid-driven fracture propagation in porous media with varying constant compatible stress boundary conditions is discussed and implemented. Since crack opening displacement, fracture path, and stress values near the fracture are highly dependent on the given boundary conditions, it is crucial to take into account the impact of in situ stresses on fracturing propagation for realistic applications. We illustrate several numerical examples that include the effects of different boundary conditions on the fracture propagation. In addition, an example using realistic boundary conditions from a reservoir simulator is included to show the capabilities of our computational framework.}
}

Summary A phase-field approach for fluid-driven fracture propagation in porous media with varying constant compatible stress boundary conditions is discussed and implemented. Since crack opening displacement, fracture path, and stress values near the fracture are highly dependent on the given boundary conditions, it is crucial to take into account the impact of in situ stresses on fracturing propagation for realistic applications. We illustrate several numerical examples that include the effects of different boundary conditions on the fracture propagation. In addition, an example using realistic boundary conditions from a reservoir simulator is included to show the capabilities of our computational framework.

@Article{LeeMinWhe2018,
author="Lee, Sanghyun
and Min, Baehyun
and Wheeler, Mary F.",
title="Optimal design of hydraulic fracturing in porous media using the phase field fracture model coupled with genetic algorithm",
journal="Computational Geosciences",
year="2018",
month="Jun",
day="01",
volume="22",
number="3",
pages="833-849",
abstract="We present a framework for the coupling of fluid-filled fracture propagation and a genetic inverse algorithm for optimizing hydraulic fracturing scenarios in porous media. Fracture propagations are described by employing a phase field approach, which treats fracture surfaces as diffusive zones rather than of interfaces. Performance of the coupled approach is provided with applications to numerical experiments related to maximizing production or reservoir history matching for emphasizing the capability of the framework.",
issn="1573-1499",
doi="10.1007/s10596-018-9728-6",
url="https://doi.org/10.1007/s10596-018-9728-6"
}

We present a framework for the coupling of fluid-filled fracture propagation and a genetic inverse algorithm for optimizing hydraulic fracturing scenarios in porous media. Fracture propagations are described by employing a phase field approach, which treats fracture surfaces as diffusive zones rather than of interfaces. Performance of the coupled approach is provided with applications to numerical experiments related to maximizing production or reservoir history matching for emphasizing the capability of the framework.

@article{LeeWhe2017_egtwo,
title = "Enriched Galerkin methods for two-phase flow in porous media with capillary pressure",
journal = "Journal of Computational Physics",
volume = "367",
pages = "65 - 86",
year = "2018",
issn = "0021-9991",
doi = "https://doi.org/10.1016/j.jcp.2018.03.031",
url = "http://www.sciencedirect.com/science/article/pii/S0021999118301918",
author = "Sanghyun Lee and Mary F. Wheeler",
keywords = "Enriched Galerkin finite element methods, Two-phase flow, Capillary pressure, Porous media, Entropy viscosity, Dynamic mesh adaptivity",
abstract = "In this paper, we propose an enriched Galerkin (EG) approximation for a two-phase pressure saturation system with capillary pressure in heterogeneous porous media. The EG methods are locally conservative, have fewer degrees of freedom compared to discontinuous Galerkin (DG), and have an efficient pressure solver. To avoid non-physical oscillations, an entropy viscosity stabilization method is employed for high order saturation approximations. Entropy residuals are applied for dynamic mesh adaptivity to reduce the computational cost for larger computational domains. The iterative and sequential IMplicit Pressure and Explicit Saturation (IMPES) algorithms are treated in time. Numerical examples with different relative permeabilities and capillary pressures are included to verify and to demonstrate the capabilities of EG."
}

In this paper, we propose an enriched Galerkin (EG) approximation for a two-phase pressure saturation system with capillary pressure in heterogeneous porous media. The EG methods are locally conservative, have fewer degrees of freedom compared to discontinuous Galerkin (DG), and have an efficient pressure solver. To avoid non-physical oscillations, an entropy viscosity stabilization method is employed for high order saturation approximations. Entropy residuals are applied for dynamic mesh adaptivity to reduce the computational cost for larger computational domains. The iterative and sequential IMplicit Pressure and Explicit Saturation (IMPES) algorithms are treated in time. Numerical examples with different relative permeabilities and capillary pressures are included to verify and to demonstrate the capabilities of EG.

@article{LeeMikWheWick2017_pftwo,
author = {Lee, S. and Mikeli\'c, A. and Wheeler, M. and Wick, T.},
title = {Phase-Field Modeling of Two Phase Fluid Filled Fractures in a Poroelastic Medium},
journal = {Multiscale Modeling \& Simulation},
volume = {16},
number = {4},
pages = {1542-1580},
year = {2018},
doi = {10.1137/17M1145239},
URL = {https://doi.org/10.1137/17M1145239},
eprint = {https://doi.org/10.1137/17M1145239}
}

@article{ChooLee_2018,
title = "Enriched Galerkin finite elements for coupled poromechanics with local mass conservation",
journal = "Computer Methods in Applied Mechanics and Engineering",
volume = "341",
pages = "311 - 332",
year = "2018",
issn = "0045-7825",
doi = "https://doi.org/10.1016/j.cma.2018.06.022",
url = "http://www.sciencedirect.com/science/article/pii/S0045782518303207",
author = "Jinhyun Choo and Sanghyun Lee",
keywords = "Enriched Galerkin method, Finite element method, Coupled poromechanics, Local mass conservation"
}

@inproceedings{choo2018enriched,
  title={Enriched Galerkin Finite Element Method for Locally Mass Conservative Simulation of Coupled Hydromechanical Problems},
  author={Choo, Jinhyun and Lee, Sanghyun},
  booktitle={Proceedings of China-Europe Conference on Geotechnical Engineering},
  pages={312--315},
  year={2018},
  organization={Springer}
}

@article{LeeWheWickSri2016,
author = "Sanghyun Lee and Mary F. Wheeler and Thomas Wick and Sanjay Srinivasan",
title = "Initialization of phase-field fracture propagation in porous media using probability maps of fracture networks. ",
journal = "Mechanics Research Communications ",
volume = "80",
number = "",
pages = "16-23",
year = "2017",
note = "Multi-Physics of Solids at Fracture",
issn = "0093-6413",
doi = "http://dx.doi.org/10.1016/j.mechrescom.2016.04.002",
url = "http://www.sciencedirect.com/science/article/pii/S0093641316300106",
keywords = {Porous media,Hydraulic fracturing, Probability map, Phase-field fracture},
abstract = "It is well known in the geophysical community that surface deflection information/micro-seismic data are considered to be one of the best diagnostics for revealing the volume of rock fracture. However, the in-exactness of the data representing the deformation induced to calibrate and represent complex fracture networks created and connected during hydraulic fracturing presents a challenge. In this paper, we propose a technique that implements a phase-field approach to propagate fractures and their interaction with existing fracture networks using surface deflection data. The latter one provides a probability map of fractures in a heterogeneous reservoir. These data are used to initialize both the location of the fractures and the phase-field function. In addition, this approach has the potential for optimizing well placement/spacing for fluid-filled fracture propagation for oil and gas production and or carbon sequestration and utilization. Using prototype models based on realistic field data, we demonstrate the effects of interactions between existing and propagating fractures in terms of several numerical simulations with different probability thresholds, locations, and numbers of fractures. Our results indicate that propagating fractures interact in a complex manner with the existing fracture network. The modeled propagation of hydraulic fractures is sensitive to the threshold employed within the phase-field approach for delineating fractures. "
}

It is well known in the geophysical community that surface deflection information/micro-seismic data are considered to be one of the best diagnostics for revealing the volume of rock fracture. However, the in-exactness of the data representing the deformation induced to calibrate and represent complex fracture networks created and connected during hydraulic fracturing presents a challenge. In this paper, we propose a technique that implements a phase-field approach to propagate fractures and their interaction with existing fracture networks using surface deflection data. The latter one provides a probability map of fractures in a heterogeneous reservoir. These data are used to initialize both the location of the fractures and the phase-field function. In addition, this approach has the potential for optimizing well placement/spacing for fluid-filled fracture propagation for oil and gas production and or carbon sequestration and utilization. Using prototype models based on realistic field data, we demonstrate the effects of interactions between existing and propagating fractures in terms of several numerical simulations with different probability thresholds, locations, and numbers of fractures. Our results indicate that propagating fractures interact in a complex manner with the existing fracture network. The modeled propagation of hydraulic fractures is sensitive to the threshold employed within the phase-field approach for delineating fractures.

@article{LeeWheWick2016_Iter,
author = "Sanghyun Lee and Mary F. Wheeler and Thomas Wick",
title = "Iterative coupling of flow, geomechanics and adaptive phase-field fracture including level-set crack width approaches",
journal = "Journal of Computational and Applied Mathematics ",
volume = "314",
number = "",
pages = "40 - 60",
year = "2017",
note = "",
issn = "0377-0427",
doi = "http://dx.doi.org/10.1016/j.cam.2016.10.022",
url = "http://www.sciencedirect.com/science/article/pii/S0377042716305118",
keywords = {Fluid-filled fracture, Phase-field fracture,Fixed stress splitting,
            Level set, Crack width, Porous media},
abstract={In this work, we present numerical studies of fixed-stress iterative coupling for solving flow and geomechanics with propagating fractures in a porous medium. Specifically, fracture propagations are described by employing a phase-field approach. The extension to fixed-stress splitting to propagating phase-field fractures and systematic investigation of its properties are important enhancements to existing studies. Moreover, we provide an accurate computation of the fracture opening using level-set approaches and a subsequent finite element interpolation of the width. The latter enters as fracture permeability into the pressure diffraction problem which is crucial for fluid filled fractures. Our developments are substantiated with several numerical tests that include comparisons of computational cost for iterative coupling and nonlinear and linear iterations as well as convergence studies in space and time.}
}

In this work, we present numerical studies of fixed-stress iterative coupling for solving flow and geomechanics with propagating fractures in a porous medium. Specifically, fracture propagations are described by employing a phase-field approach. The extension to fixed-stress splitting to propagating phase-field fractures and systematic investigation of its properties are important enhancements to existing studies. Moreover, we provide an accurate computation of the fracture opening using level-set approaches and a subsequent finite element interpolation of the width. The latter enters as fracture permeability into the pressure diffraction problem which is crucial for fluid filled fractures. Our developments are substantiated with several numerical tests that include comparisons of computational cost for iterative coupling and nonlinear and linear iterations as well as convergence studies in space and time.

@article{LeeWhe2016_EG,
title = "Adaptive enriched Galerkin methods for miscible displacement problems with entropy residual stabilization",
journal = "Journal of Computational Physics ",
volume = "331",
number = "",
pages = "19 - 37",
year = "2017",
note = "",
issn = "0021-9991",
doi = "http://dx.doi.org/10.1016/j.jcp.2016.10.072",
url = "http://www.sciencedirect.com/science/article/pii/S0021999116305952",
author = "Sanghyun Lee and Mary F. Wheeler",
keywords = {Porous media, Enriched Galerkin,
Miscible displacement,
Viscous fingering,
Locally conservative,
Entropy viscosity,
Hele-Shaw,
Adaptive finite elements}, 
abstract = "We present a novel approach to the simulation of miscible displacement by employing adaptive enriched Galerkin finite element methods (EG) coupled with entropy residual stabilization for transport. In particular, numerical simulations of viscous fingering instabilities in heterogeneous porous media and Hele–Shaw cells are illustrated. EG is formulated by enriching the conforming continuous Galerkin finite element method (CG) with piecewise constant functions. The method provides locally and globally conservative fluxes, which are crucial for coupled flow and transport problems. Moreover, EG has fewer degrees of freedom in comparison with discontinuous Galerkin (DG) and an efficient flow solver has been derived which allows for higher order schemes. Dynamic adaptive mesh refinement is applied in order to reduce computational costs for large-scale three dimensional applications. In addition, entropy residual based stabilization for high order EG transport systems prevents spurious oscillations. Numerical tests are presented to show the capabilities of EG applied to flow and transport."
}

We present a novel approach to the simulation of miscible displacement by employing adaptive enriched Galerkin finite element methods (EG) coupled with entropy residual stabilization for transport. In particular, numerical simulations of viscous fingering instabilities in heterogeneous porous media and Hele–Shaw cells are illustrated. EG is formulated by enriching the conforming continuous Galerkin finite element method (CG) with piecewise constant functions. The method provides locally and globally conservative fluxes, which are crucial for coupled flow and transport problems. Moreover, EG has fewer degrees of freedom in comparison with discontinuous Galerkin (DG) and an efficient flow solver has been derived which allows for higher order schemes. Dynamic adaptive mesh refinement is applied in order to reduce computational costs for large-scale three dimensional applications. In addition, entropy residual based stabilization for high order EG transport systems prevents spurious oscillations. Numerical tests are presented to show the capabilities of EG applied to flow and transport.

@article{ScoWheMikLee2017_JCP,
author = "Guglielmo Scovazzi and Mary F. Wheeler and Andro Mikeli{\'c} and Sanghyun Lee",
title = "Analytical and variational numerical methods for unstable miscible displacement flows in porous media",
journal = "Journal of Computational Physics ",
volume = "335",
number = "",
pages = "444 - 496",
year = "2017",
note = "",
issn = "0021-9991",
doi = "http://doi.org/10.1016/j.jcp.2017.01.021",
url = "http://www.sciencedirect.com/science/article/pii/S0021999117300311",
keywords = {Porous media, Miscible displacement,
Viscous fingering,
Hele-Shaw}, 
abstract ="Analytical and variational numerical methods for unstable miscible displacement flows in porous media."
}

Analytical and variational numerical methods for unstable miscible displacement flows in porous media.

@proceedings{AlmLeeWheelerWick2017_RSC,
  title = {Multirate Coupling for Flow and Geomechanics Applied to Hydraulic Fracturing Using an Adaptive Phase-Field Technique},
  author={Tameem Almani and Sanghyun Lee and  Mary F. Wheeler and  Thomas Wick},
  year         = {2017},
  organization = {Society of Petroleum Engineers},
  publisher    = {SPE Reservoir Simulation Conference},  
  volume = {SPE-182610-MS},
  keywords = {Porous media,Fluid-filled fracture, Phase-field fracture, Fixed stress splitting}, 
  abstract = "We present and analyze a multirate fixed stress split iterative coupling scheme for coupling flow and geomechanics in a poroelastic medium involving fracture propagation modeled with a phase field approach. The novelty of this work lies in the efficient integration of the fixed-stress split coupling scheme with phase-field fracture propagation models. The multirate coupling algorithm utilizes different time-scales of the flow and mechanics problems, by allowing for multiple finer time steps for flow within one coarse mechanics time step. When applied to production scenarios, the multirate scheme results in massive reductions in the number of mechanics linear iterations, without jeopardizing the accuracy of the obtained results. A number of numerical simulations substantiate our algorithmic developments. These tests include prototype computations, multiple propagating fractures, and fractures initialized by a microseismic probability map."
  }

We present and analyze a multirate fixed stress split iterative coupling scheme for coupling flow and geomechanics in a poroelastic medium involving fracture propagation modeled with a phase field approach. The novelty of this work lies in the efficient integration of the fixed-stress split coupling scheme with phase-field fracture propagation models. The multirate coupling algorithm utilizes different time-scales of the flow and mechanics problems, by allowing for multiple finer time steps for flow within one coarse mechanics time step. When applied to production scenarios, the multirate scheme results in massive reductions in the number of mechanics linear iterations, without jeopardizing the accuracy of the obtained results. A number of numerical simulations substantiate our algorithmic developments. These tests include prototype computations, multiple propagating fractures, and fractures initialized by a microseismic probability map.

@article {BonGueLee2016,
author = {Bonito, Andrea and Guermond, Jean-Luc and Lee, Sanghyun},
title = {Numerical simulations of bouncing jets},
journal = {International Journal for Numerical Methods in Fluids},
volume = {80},
number = {1},
issn = {1097-0363},
url = {http://dx.doi.org/10.1002/fld.4071},
doi = {10.1002/fld.4071},
pages = {53-75},
keywords = {Bouncing jet, Kaye effect, Entropy viscosity, Level set, Projection method, Shear-thinning viscosity, 
            Adaptive finite elements, Navier-Stokes,Multi phase flow},
year = {2016},
abstract = {Bouncing jets are fascinating phenomenon occurring under certain conditions when a jet impinges on a free surface. This effect is observed when the fluid is Newtonian and the jet falls in a bath undergoing a solid motion. It occurs also for non-Newtonian fluids when the jets fall in a vessel at rest containing the same fluid. We investigate numerically the impact of the experimental setting and the rheological properties of the fluid on the onset of the bouncing phenomenon. Our investigations show that the occurrence of a thin lubricating layer of air separating the jet and the rest of the liquid is a key factor for the bouncing of the jet to happen. The numerical technique that is used consists of a projection method for the Navier-Stokes system coupled with a level set formulation for the representation of the interface. The space approximation is carried out with adaptive finite elements. Adaptive refinement is shown to be very important to capture the thin layer of air that is responsible for the bouncing.}
}

Bouncing jets are fascinating phenomenon occurring under certain conditions when a jet impinges on a free surface. This effect is observed when the fluid is Newtonian and the jet falls in a bath undergoing a solid motion. It occurs also for non-Newtonian fluids when the jets fall in a vessel at rest containing the same fluid. We investigate numerically the impact of the experimental setting and the rheological properties of the fluid on the onset of the bouncing phenomenon. Our investigations show that the occurrence of a thin lubricating layer of air separating the jet and the rest of the liquid is a key factor for the bouncing of the jet to happen. The numerical technique that is used consists of a projection method for the Navier-Stokes system coupled with a level set formulation for the representation of the interface. The space approximation is carried out with adaptive finite elements. Adaptive refinement is shown to be very important to capture the thin layer of air that is responsible for the bouncing.

@article{LeeMikWheWick2016_Prop,
author = "Sanghyun Lee and Andro Mikeli{\'c}  and Mary F. Wheeler and Thomas Wick",
title = "Phase-field modeling of proppant-filled fractures in a poroelastic medium ",
journal = "Computer Methods in Applied Mechanics and Engineering ",
volume = "312",
number = "",
pages = "509 - 541",
year = "2016",
note = "Phase Field Approaches to Fracture ",
issn = "0045-7825",
doi = "http://dx.doi.org/10.1016/j.cma.2016.02.008",
url = "http://www.sciencedirect.com/science/article/pii/S0045782516300305",
keywords = {Phase-field fracture, Hydraulic fracturing, Proppant transport,Locally conservative,
            Quasi-Newtonian flow, Porous media},
abstract={In this paper we present a phase field model for proppant-filled fractures in a poroelastic medium. The formulation of the coupled system involves four unknowns: displacements, phase field, pressure, and proppant concentration. The two-field displacement phase-field system is solved fully-coupled and accounts for crack irreversibility. This solution is then coupled to the pressure equation via a fixed-stress iteration. The pressure is obtained by using a diffraction equation where the phase-field variable serves as an indicator function that distinguishes between the fracture and the reservoir. The transport of the proppant in the fracture is modeled by using a power-law fluid system. The numerical discretization in space is based on Galerkin finite elements for displacements and phase-field, and an enriched Galerkin method is applied for the pressure equation in order to obtain local mass conservation. The concentration is solved with cell-centered finite elements. Nonlinear equations are treated with Newton’s method. Our developments are substantiated with several numerical examples in two and three dimensions.}
}

In this paper we present a phase field model for proppant-filled fractures in a poroelastic medium. The formulation of the coupled system involves four unknowns: displacements, phase field, pressure, and proppant concentration. The two-field displacement phase-field system is solved fully-coupled and accounts for crack irreversibility. This solution is then coupled to the pressure equation via a fixed-stress iteration. The pressure is obtained by using a diffraction equation where the phase-field variable serves as an indicator function that distinguishes between the fracture and the reservoir. The transport of the proppant in the fracture is modeled by using a power-law fluid system. The numerical discretization in space is based on Galerkin finite elements for displacements and phase-field, and an enriched Galerkin method is applied for the pressure equation in order to obtain local mass conservation. The concentration is solved with cell-centered finite elements. Nonlinear equations are treated with Newton’s method. Our developments are substantiated with several numerical examples in two and three dimensions.

@article{LeeWheWick2016_Pf,
author = "Sanghyun Lee and Mary F. Wheeler and Thomas Wick",
title = "Pressure and fluid-driven fracture propagation in porous media using an adaptive finite element phase field model ",
journal = "Computer Methods in Applied Mechanics and Engineering ",
volume = "305",
number = "",
pages = "111 - 132",
year = "2016",
note = "",
issn = "0045-7825",
doi = "http://dx.doi.org/10.1016/j.cma.2016.02.037",
url = "http://www.sciencedirect.com/science/article/pii/S0045782516300676",
keywords = {Phase-field fracture, Fluid-filled fracture, Hydraulic fracturing, 
            Adaptive finite elements, Porous media},
abstract = "Abstract This work presents phase field fracture modeling in heterogeneous porous media. We develop robust and efficient numerical algorithms for pressure-driven and fluid-driven settings in which the focus relies on mesh adaptivity in order to save computational cost for large-scale 3D applications. In the fluid-driven framework, we solve for three unknowns pressure, displacements and phase field that are treated with a fixed-stress iteration in which the pressure and the displacemet-phase-field system are decoupled. The latter subsystem is solved with a combined Newton approach employing a primal-dual active set method in order to account for crack irreversibility. Numerical examples for pressurized fractures and fluid filled fracture propagation in heterogeneous porous media demonstrate our developments. In particular, mesh refinement allows us to perform systematic studies with respect to the spatial discretization parameter. "
}

Abstract This work presents phase field fracture modeling in heterogeneous porous media. We develop robust and efficient numerical algorithms for pressure-driven and fluid-driven settings in which the focus relies on mesh adaptivity in order to save computational cost for large-scale 3D applications. In the fluid-driven framework, we solve for three unknowns pressure, displacements and phase field that are treated with a fixed-stress iteration in which the pressure and the displacemet-phase-field system are decoupled. The latter subsystem is solved with a combined Newton approach employing a primal-dual active set method in order to account for crack irreversibility. Numerical examples for pressurized fractures and fluid filled fracture propagation in heterogeneous porous media demonstrate our developments. In particular, mesh refinement allows us to perform systematic studies with respect to the spatial discretization parameter.

@article{LeeLeeWhe2016_EG,
author = {Sanghyun Lee and Young-Ju Lee and Mary F. Wheeler},
title = {A Locally Conservative Enriched Galerkin Approximation and Efficient Solver for Elliptic and Parabolic Problems},
journal = {SIAM Journal on Scientific Computing},
volume = {38},
number = {3},
pages = {A1404-A1429},
year = {2016},
doi = {10.1137/15M1041109},
URL = {http://dx.doi.org/10.1137/15M1041109},
keywords={Porous media,Enriched Galerkin, Locally conservative, Solver}, 
abstract={We present and analyze an enriched Galerkin finite element method (EG) to solve elliptic and parabolic equations with jump coefficients. EG is formulated by enriching the conforming continuous Galerkin finite element method (CG) with piecewise constant functions which can be considered as a penalty stabilization. The method is shown to be locally and globally conservative, while keeping fewer degrees of freedom in comparison with discontinuous Galerkin finite element methods (DG). Moreover, we present and analyze a fast effective EG solver whose cost is roughly that of CG and which can handle an arbitrary order of approximations. A number of numerical tests in two and three dimensions are presented to confirm our theoretical results as well as to demonstrate the advantages of EG when coupled with transport.}
}

We present and analyze an enriched Galerkin finite element method (EG) to solve elliptic and parabolic equations with jump coefficients. EG is formulated by enriching the conforming continuous Galerkin finite element method (CG) with piecewise constant functions which can be considered as a penalty stabilization. The method is shown to be locally and globally conservative, while keeping fewer degrees of freedom in comparison with discontinuous Galerkin finite element methods (DG). Moreover, we present and analyze a fast effective EG solver whose cost is roughly that of CG and which can handle an arbitrary order of approximations. A number of numerical tests in two and three dimensions are presented to confirm our theoretical results as well as to demonstrate the advantages of EG when coupled with transport.

@article{LeeSal2016,
title = "Stability analysis of pressure correction schemes for the Navier-Stokes equations with traction boundary conditions ",
journal = "Computer Methods in Applied Mechanics and Engineering ",
volume = "309",
number = "",
pages = "307 - 324",
year = "2016",
note = "",
issn = "0045-7825",
doi = "http://dx.doi.org/10.1016/j.cma.2016.05.043",
url = "http://www.sciencedirect.com/science/article/pii/S0045782516304923",
author = "Sanghyun Lee and Abner J. Salgado",
keywords = {Projection method, Open and traction boundary conditions,
            Fractional time stepping, Navier-Stokes},
abstract = "We present a stability analysis for two different rotational pressure correction schemes with open and traction boundary conditions. First, we provide a stability analysis for a rotational version of the grad-div stabilized scheme of Bonito et al. (2015). This scheme turns out to be unconditionally stable, provided the stabilization parameter is suitably chosen. We also establish a conditional stability result for the boundary correction scheme presented in Bansch (2014). These results are shown by employing the equivalence between stabilized gauge Uzawa methods and rotational pressure correction schemes with traction boundary conditions. "
}

We present a stability analysis for two different rotational pressure correction schemes with open and traction boundary conditions. First, we provide a stability analysis for a rotational version of the grad-div stabilized scheme of Bonito et al. (2015). This scheme turns out to be unconditionally stable, provided the stabilization parameter is suitably chosen. We also establish a conditional stability result for the boundary correction scheme presented in Bansch (2014). These results are shown by employing the equivalence between stabilized gauge Uzawa methods and rotational pressure correction schemes with traction boundary conditions.

@article{LeeRebHayWhe_2016,
author = {Lee, Sanghyun and Reber, Jacqueline E. and Hayman, Nicholas W. and Wheeler, Mary F.},
title = {Investigation of wing crack formation with a combined phase-field and experimental approach},
journal = {Geophysical Research Letters},
volume = {43},
number = {15},
issn = {1944-8007},
url = {http://dx.doi.org/10.1002/2016GL069979},
doi = {10.1002/2016GL069979},
pages = {7946--7952},
keywords = {Phase-field fracture, Wing crack, Experiments},
year = {2016},
abstract={Fractures that propagate off of weak slip planes are known as wing cracks and often play important roles in both tectonic deformation and fluid flow across reservoir seals. Previous numerical models have produced the basic kinematics of wing crack openings but generally have not been able to capture fracture geometries seen in nature. Here we present both a phase-field modeling approach and a physical experiment using gelatin for a wing crack formation. By treating the fracture surfaces as diffusive zones instead of as discontinuities, the phase-field model does not require consideration of unpredictable rock properties or stress inhomogeneities around crack tips. It is shown by benchmarking the models with physical experiments that the numerical assumptions in the phase-field approach do not affect the final model predictions of wing crack nucleation and growth. With this study, we demonstrate that it is feasible to implement the formation of wing cracks in large scale phase-field reservoir models.}
}

Fractures that propagate off of weak slip planes are known as wing cracks and often play important roles in both tectonic deformation and fluid flow across reservoir seals. Previous numerical models have produced the basic kinematics of wing crack openings but generally have not been able to capture fracture geometries seen in nature. Here we present both a phase-field modeling approach and a physical experiment using gelatin for a wing crack formation. By treating the fracture surfaces as diffusive zones instead of as discontinuities, the phase-field model does not require consideration of unpredictable rock properties or stress inhomogeneities around crack tips. It is shown by benchmarking the models with physical experiments that the numerical assumptions in the phase-field approach do not affect the final model predictions of wing crack nucleation and growth. With this study, we demonstrate that it is feasible to implement the formation of wing cracks in large scale phase-field reservoir models.

@article{BonGueLee2015,
author="Bonito, Andrea
and Guermond, Jean-Luc
and Lee, Sanghyun",
editor="Abdulle, Assyr
and Deparis, Simone
and Kressner, Daniel
and Nobile, Fabio
and Picasso, Marco",
title="Modified Pressure-Correction Projection Methods: Open Boundary and Variable Time Stepping",
journal="Numerical Mathematics and Advanced  Applications - ENUMATH 2013: Proceedings of ENUMATH 2013, the 10th European Conference on Numerical Mathematics and Advanced Applications, Lausanne, August 2013",
year="2015",
publisher="Springer International Publishing",
pages="623-631",
isbn="978-3-319-10705-9",
doi="10.1007/978-3-319-10705-9_61",
url="http://dx.doi.org/10.1007/978-3-319-10705-9_61",
keywords={Projection method, Variable time stepping, Navier-Stokes},
abstract={In this paper, we design and study two modifications of the first order
standard pressure increment projection scheme for the Stokes system. The first
scheme improves the existing schemes in the case of open boundary condition
by modifying the pressure increment boundary condition, thereby minimizing the
pressure boundary layer and recovering the optimal first order decay. The second
scheme allows for variable time stepping. It turns out that the straightforward
modification to variable time stepping leads to unstable schemes. The proposed
scheme is not only stable but also exhibits the optimal first order decay. Numerical
computations illustrating the theoretical estimates are provided for both new
schemes}
}

In this paper, we design and study two modifications of the first order standard pressure increment projection scheme for the Stokes system. The first scheme improves the existing schemes in the case of open boundary condition by modifying the pressure increment boundary condition, thereby minimizing the pressure boundary layer and recovering the optimal first order decay. The second scheme allows for variable time stepping. It turns out that the straightforward modification to variable time stepping leads to unstable schemes. The proposed scheme is not only stable but also exhibits the optimal first order decay. Numerical computations illustrating the theoretical estimates are provided for both new schemes

@proceedings{WickLeeWhe2015_3D,
  title={3D Phase-Field for Pressurized Fracture Propagation in Heterogeneous Media},
  author={Wick, Thomas and Lee, Sanghyun and Wheeler, Mary F},
  booktitle={Coupled Problems in Science and Engineering VI},
  year={2015},
  pages={605--613},
  editor={B. Schrefler, E. Oñate and M. Papadrakakis},
  publisher={International Center for Numerical Methods in Engineering (CIMNE) Barcelona, Spain},
  abstract={This work presents recent progress in phase-field-based fracture modeling in
heterogeneous porous media. Existing algorithms that have been developed in the last
years are extended to tackle three-dimensional configurations. Our solution technique
is formulated in terms of a nested Newton loop that combines a primal-dual active set
method (required for treating the crack irreversibility) and a Newton method to solve
the nonlinear, fully-coupled PDE system. An advanced numerical test demonstrates the
capabilities of our method.},
  keywords={Phase-field fracture, Porous media, Adaptive finite elements}
}

This work presents recent progress in phase-field-based fracture modeling in heterogeneous porous media. Existing algorithms that have been developed in the last years are extended to tackle three-dimensional configurations. Our solution technique is formulated in terms of a nested Newton loop that combines a primal-dual active set method (required for treating the crack irreversibility) and a Newton method to solve the nonlinear, fully-coupled PDE system. An advanced numerical test demonstrates the capabilities of our method.

@phdthesis{Lee2014_Phd,
author={Lee,Sanghyun},
year={2014},
title={Numerical simulations of bouncing jets},
school={Texas A\&M University},
pages={115},
abstract={The Kaye effect is a fascinating phenomenon of a leaping shampoo stream which was first described by Alan Kaye in 1963 as a property of non-Newtonian fluid. It manifest itself when a thin stream of non-Newtonian fluid is poured into a dish of fluid. As pouring proceeds, a small stream of liquid occasionally leaps upward from the heap. We investigate numerically the impact of the experimental setting as well as the fluid rheology on the apparition of bouncing jets. In particular, we observe the importance of the creation of a thin lubricating layer of air between the jet and the rest of the liquid. The numerical method consists of a projection method coupled with a level set formulation for the interface representation. Adaptive finite element methods are advocated to capture the different length scales inherent to this context. In addition, we design and study two modifications of the first order standard pressure correction projection scheme for the Stokes system. The first scheme improves the existing schemes in the case of open boundary condition by modifying the pressure increment boundary condition, thereby minimizing the pressure boundary layer and recovering the optimal first order decay. The second scheme allows for variable time stepping. It turns out that the straightforward modification to variable time stepping leads to unstable schemes. The proposed scheme is not only stable but also exhibits the optimal first order decay. Numerical computations illustrating the theoretical estimates are provided for both 
new schemes.},
keywords={Bouncing jet, 
Entropy viscosity, 
Navier-Stokes,
Kaye effect, 
Level set, 
Projection method, 
Reinitialization,
Multi phase flow, 
Variable time stepping},
isbn={9781321586541},
language={English},
url={http://ezproxy.lib.utexas.edu/login?url=http://search.proquest.com/docview/1664841685?accountid=7118}
}

The Kaye effect is a fascinating phenomenon of a leaping shampoo stream which was first described by Alan Kaye in 1963 as a property of non-Newtonian fluid. It manifest itself when a thin stream of non-Newtonian fluid is poured into a dish of fluid. As pouring proceeds, a small stream of liquid occasionally leaps upward from the heap. We investigate numerically the impact of the experimental setting as well as the fluid rheology on the apparition of bouncing jets. In particular, we observe the importance of the creation of a thin lubricating layer of air between the jet and the rest of the liquid. The numerical method consists of a projection method coupled with a level set formulation for the interface representation. Adaptive finite element methods are advocated to capture the different length scales inherent to this context. In addition, we design and study two modifications of the first order standard pressure correction projection scheme for the Stokes system. The first scheme improves the existing schemes in the case of open boundary condition by modifying the pressure increment boundary condition, thereby minimizing the pressure boundary layer and recovering the optimal first order decay. The second scheme allows for variable time stepping. It turns out that the straightforward modification to variable time stepping leads to unstable schemes. The proposed scheme is not only stable but also exhibits the optimal first order decay. Numerical computations illustrating the theoretical estimates are provided for both new schemes.

@article{Lee2014_PRE,
  title = {Leaping shampoo glides on a lubricating air layer},
  author = {Lee, Sanghyun. and Li, E. Q. and Marston, J. O. and Bonito, Andrea. and Thoroddsen, S. T.},
  journal = {Phys. Rev. E},
  volume = {87},
  issue = {6},
  pages = {061001},
  numpages = {4},
  year = {2013},
  month = {Jun},
  publisher = {American Physical Society},
  doi = {10.1103/PhysRevE.87.061001},
  url = {http://link.aps.org/doi/10.1103/PhysRevE.87.061001},
  keywords={Bouncing jet, Kaye effect, Experiments}, 
  abstract={When a stream of shampoo is fed onto a pool in one's hand, a jet can leap sideways or rebound from the liquid surface in an intriguing phenomenon known as the Kaye effect. Earlier studies have debated whether non-Newtonian effects are the underlying cause of this phenomenon, making the jet glide on top of a shear-thinning liquid layer, or whether an entrained air layer is responsible. Herein we show unambiguously that the jet slides on a lubricating air layer. We identify this layer by looking through the pool liquid and observing its rupture into fine bubbles. The resulting microbubble sizes suggest this air layer is of submicron thickness. This thickness estimate is also supported by the tangential deceleration of the jet during the rebounding.}
}

When a stream of shampoo is fed onto a pool in one's hand, a jet can leap sideways or rebound from the liquid surface in an intriguing phenomenon known as the Kaye effect. Earlier studies have debated whether non-Newtonian effects are the underlying cause of this phenomenon, making the jet glide on top of a shear-thinning liquid layer, or whether an entrained air layer is responsible. Herein we show unambiguously that the jet slides on a lubricating air layer. We identify this layer by looking through the pool liquid and observing its rupture into fine bubbles. The resulting microbubble sizes suggest this air layer is of submicron thickness. This thickness estimate is also supported by the tangential deceleration of the jet during the rebounding.

@unpublished{LeeLeeWhe2017_egcoupled,
author = {Sanghyun Lee and Young-Ju Lee and Mary F. Wheeler},
title = {Enriched {G}alerkin approximations for coupled flow and 
transport system},
year = {In preparation}, 
keywords = {Porous media, Miscible displacement, Enriched Galerkin}
}