%%% =======================================================
%%% BibTeX-file{
%%%     author          = "James Quinlan",
%%%     version         = "1.02",
%%%     date            = "28 Sept 2021",
%%%     filename        = "vcmp.bib",
%%%     keywords        = "upscale, control-volume",
%%%     supported       = "no",
%%%     docstring       = "BibTeX bibliography for the article
%%%                        \cite{quinlan2021}:
%%%                        VCMP3D"
%%% 	AMS		= "76-S05, 76-M12, 65-N08, 86-08"
%%%     }
%%% =======================================================


% ----------------- A ------------------ %
@article{aarnes2004use,
	Author = {Aarnes, Jorg E},
	Journal = {Multiscale Modeling \& Simulation},
	Annote = {},
	Keywords = {FEM},
	Number = {3},
	Pages = {421--439},
	Publisher = {SIAM},
	Title = {On the use of a mixed multiscale finite element method for greater flexibility and increased speed or improved accuracy in reservoir simulation},
	Volume = {2},
	Year = {2004}}



@incollection{aarnes2007introduction,
  title={An introduction to the numerics of flow in porous media using {MATLAB}},
  author={Aarnes, J{\o}rg E and Gimse, Tore and Lie, Knut-Andreas},
  booktitle={Geometric modelling, numerical simulation, and optimization},
  pages={265--306},
  year={2007},
  publisher={Springer}
}




@incollection{aarnes2007modelling,
  title={Modelling of multiscale structures in flow simulations for petroleum reservoirs},
  author={Aarnes, J{\o}rg E and Kippe, Vegard and Lie, Knut-Andreas and Rustad, Alf Birger},
  booktitle={Geometric Modelling, Numerical Simulation, and Optimization},
  pages={307--360},
  year={2007},
  publisher={Springer}
}





% AMS
@Misc{AMSMSC2010,
  title =	 {{Mathematics Subject Classification}},
  author =	 {{American Mathematical Society}},
  year =	 {2010},
  url =		 {http://www.ams.org/mathscinet/msc/msc2010.html},
  urldate =	 {2015/03/29},
}




% Aavatsmark
@inproceedings{aavatsmark2007multipoint,
  title={Multipoint flux approximation methods for quadrilateral grids},
  author={Aavatsmark, Ivar},
  booktitle={9th International forum on reservoir simulation, Abu Dhabi},
  pages={9--13},
  year={2007}, 
  annote={}, 
  abstract={}, 
  apa={}, 
  keywords={}, 
  ams={}
}

% Aavatsmark
@inproceedings{aavatsmark2008comparison,
  title={Comparison of monotonicity for some multipoint flux approximation methods},
  author={Aavatsmark, Ivar},
  booktitle={Finite Volumes for Complex Applications},
  volume={5},
  pages={19--34},
  year={2008},
  organization={5, Wiley-ISTE New York}, 
  annote={}, 
  abstract={}, 
  apa={}, 
  keywords={}, 
  ams={}
}

@article{aavatsmark2001control,
	Author = {Aavatsmark, Ivar and Reiso, Edel and Teigland, Rune},
	Journal = {Computational Geosciences},
	Keywords = {quadrilateral grids},
	Number = {1},
	Pages = {1--23},
	Publisher = {Springer},
	Title = {Control-volume discretization method for quadrilateral grids with faults and local refinements},
	Volume = {5},
	Year = {2001},
	Apa = {Aavatsmark, I., Reiso, E., \& Teigland, R. (2001). Control-volume discretization method for quadrilateral grids with faults and local refinements. Computational Geosciences, 5(1), 1-23.}}



% Aavatsmark
@article{aavatsmark2007convergence,
  title={Convergence of a symmetric {MPFA} method on quadrilateral grids},
  author={Aavatsmark, Ivar and Eigestad, Geirte T and Klausen, Runhildk A and Wheeler, Mary F and Yotov, I},
  journal={Computational geosciences},
  volume={11},
  number={4},
  pages={333--345},
  year={2007},
  publisher={Springer}, 
  annote={}, 
  abstract={}, 
  apa={Aavatsmark, I., Eigestad, G. T., Klausen, R. A., Wheeler, M. F., \& Yotov, I. (2007). Convergence of a symmetric {MPFA} method on quadrilateral grids. Computational geosciences, 11(4), 333-345.}, 
  keywords={}, 
  ams={}
}



% Aavatsmark
@article{aavatsmark1996discretization,
	Annote = {Two classes of discretization (O- and U-methods) are proposed for control volume formulations on non-orthogonal (quadrilateral) grids in 2D.  \\
  \textbf{discretization non-orthogonal grids; quadrilateral grids, Curvilinear grids, anisotropic, inhomogeneous medium.} },
	Author = {Aavatsmark, Ivar and Barkve, T and B{\o}e, {\O} and Mannseth, T},
	Journal = {Journal of computational physics},
	Keywords = {discretization non-orthogonal grids; quadrilateral grids, Curvilinear grids, anisotropic, inhomogeneous medium},
	Number = {1},
	Pages = {2--14},
	Publisher = {Elsevier},
	Title = {Discretization on non-orthogonal, quadrilateral grids for inhomogeneous anisotropic media},
	Volume = {127},
	Year = {1996}}



% Aavatsmark
@article{aavatsmark2007interpretation,
  title={Interpretation of a two-point flux stencil for skew parallelogram grids},
  author={Aavatsmark, Ivar},
  journal={Computational geosciences},
  volume={11},
  number={3},
  pages={199--206},
  year={2007},
  publisher={Springer}, 
  keywords={}, 
  annote={}
}


% Aavatsmark
@article{aavatsmark2002introduction,
  title={An introduction to multipoint flux approximations for quadrilateral grids},
  author={Aavatsmark, Ivar},
  journal={Computational Geosciences},
  volume={6},
  number={3-4},
  pages={405--432},
  year={2002},
  publisher={Springer}, 
  keywords={MPFA, anisotropy, inhomogeneity },
  annote={Control-volume discretizations using multipoint flux approximations (MPFA) were developed in the 1990s. This paper introduces these methods for quadrilateral grids in two and three dimensions. The introduction is kept on a basic level and gives a summary of more advanced results. Only the O-method with surface midpoints as continuity points is discussed. Flux expressions are derived both in physical and in curvilinear space. Equations for calculating the transmissibility coefficients are given, and an explicit solution is shown for constant coefficients. K-orthogonality, stability, and monotonicity are discussed, and an iterative solution technique is presented. Two numerical examples close the paper.}
}


% Aavatsmark
@article{aavatsmark2006numerical,
  title={Numerical convergence of the {MPFA} {O}-method and {U}-method for general quadrilateral grids},
  author={Aavatsmark, I and Eigestad, GT},
  journal={International journal for numerical methods in fluids},
  volume={51},
  number={9-10},
  pages={939--961},
  year={2006},
  publisher={Wiley Online Library}, 
  annote={}, 
  abstract={}, 
  apa={}, 
  keywords={MPFA, convergence}, 
  ams={}
}


@incollection{aavatsmark2006numericalbook,
  title={Numerical convergence of the MPFA O-method for general quadrilateral grids in two and three dimensions},
  author={Aavatsmark, Ivar and Eigestad, Geir Terje and Klausen, Runhild Aae},
  booktitle={Compatible spatial discretizations},
  pages={1--21},
  year={2006},
  publisher={Springer} ,
  apa={Aavatsmark, I., Eigestad, G. T., \& Klausen, R. A. (2006). Numerical convergence of the MPFA O-method for general quadrilateral grids in two and three dimensions. In \textit{Compatible spatial discretizations} (pp. 1-21). Springer, New York, NY.}
}


% Aavatsmark
@article{aavatsmark1998discretization,
  title={Discretization on unstructured grids for inhomogeneous, anisotropic media. Part I: Derivation of the methods},
  author={Aavatsmark, Ivar and Barkve, Tor and B{\o}e, O and Mannseth, Trond},
  journal={SIAM Journal on Scientific Computing},
  volume={19},
  number={5},
  pages={1700--1716},
  year={1998},
  publisher={SIAM}, 
  Annote={Discretization methods are proposed for control-volume formulations on polygonal and triangular grid cells in two space dimensions. The methods apply to any system of conservation laws where the flow density is defined by a gradient law, like Darcy's law for porous-media flow. A strong feature of the methods is the ability to handle media inhomogeneities in combination with full-tensor anisotropy. This paper gives a derivation of the methods, and the relation to previously published methods is also discussed}
}







% Aavatsmark 2008
@article{aavatsmark2008compact,
	Author = {Aavatsmark, I and Eigestad, GT and Mallison, BT and Nordbotten, JM},
	Journal = {Numerical Methods for Partial Differential Equations},
	Keywords = {MPFA; L-method; O-method; control-volume method; monotonicity},
	Number = {5},
	Pages = {1329--1360},
	Publisher = {Wiley Online Library},
	Title = {A compact multipoint flux approximation method with improved robustness},
	Volume = {24},
	Year = {2008},
	Annote = {Cited 77 times, Multipoint flux approximation (MPFA) methods were introduced to solve control-volume formulations on general grids. MPFA  convergence and monotonicity properties vary. We introduce a new MPFA method for quadrilateral grids termed the \hl{L-method}, which seeks to minimize the number of entries in the flux stencils while honoring uniform flow fields and is valid for general media. For homogeneous media and uniform grids in two dimensions, this method has four-point flux stencils and seven-point cell stencils, whereas the MPFA \hl{O-methods} has six-point flux stencils and nine-point cell stencils. The reduced stencil of the L-method appears as a consequence of adapting the method to the closest neighboring cells or, equivalently, to the dominating principal direction of anisotropy. We have tested this method's convergence and monotonicity properties and compared it with the O-methods. For moderate grids, the convergence rates are the same, but for rough grids with large aspect ratios, the convergence of the O-methods is lost, while the L-method converges with a reduced convergence rate. Also, the L-method has a larger monotonicity range than the O-methods. For homogeneous media and uniform parallelogram grids, the matrix of coefficients is an M-matrix whenever the method is monotone. For strongly nonmonotone cases, the oscillations observed for the O-methods are almost removed for the L-method. Instead, extrema on no-flow boundaries is observed. These undesired solutions, which only occur for parameters not common in applications, should be avoided by requiring that the previously derived monotonicity conditions are satisfied. For local grid refinements, test runs indicate that the L-method yields almost optimal solutions and is considerably better than those obtained by the O-methods. The efficiency of the linear solver is, in many cases, better for the L-method than for the O-methods. This is due to lower condition numbers and fewer entries in the matrix of coefficients.}}






% Amirsardari: GRIDDING
@article{amirsardari2015anisotropic,
  title={Anisotropic grid generation for dynamic simulation in heterogeneous hydrocarbon reservoirs},
  author={Amirsardari, Mahdi and Dabir, Bahram and Naderifar, Abbas},
  journal={Journal of Natural Gas Science and Engineering},
  volume={27},
  pages={1523--1535},
  year={2015},
  publisher={Elsevier}, 
  keywords={Unstructured gridding, Reservoir simulation, Anisotropic grid, Porous media},
  Annote ={Coarsening from fine geological model to coarse dynamic model is one of the main practical stages in reservoir simulation process in order to minimize simulation run time. In order to preserve accuracy and minimize simulation run time simultaneously, non-uniform element distribution is required to capture dynamic interest and high-flow regions in heterogeneous porous media.}
}



% Arbogast 2003
@article{arbogast2003overview,
	Author = {Arbogast, Todd},
	Date-Modified = {2019-01-17 14:06:33 -0500},
	Journal = {Contemporary Mathematics},
	Keywords = {FEM, upscaling elliptic problems},
	Pages = {21--32},
	Publisher = {Providence, RI: American Mathematical Society},
	Title = {An overview of subgrid upscaling for elliptic problems in mixed form},
	Volume = {329},
	Year = {2003},
	Abstract = {Place abstract here}}



% Aziz 1979 
@book{aziz1979petroleum,
	Author = {Aziz, Khalid and Settari, Antonin},
	Keywords = {TPFA},
	Publisher = {Chapman \& Hall},
	Title = {Petroleum reservoir simulation},
	Year = {1979}}
% ------------------------------------ %
















% ----------------- B ------------------ %


% Bear
@book{bear2013dynamics,
  title={Dynamics of fluids in porous media},
  author={Bear, Jacob},
  year={2013},
  publisher={Courier Corporation}, 
  keywords={Books}, 
  annote={1972 Dover Book}
}



%  Berger 1989 (Gridding)
@article{berger1989local,
	Author = {Berger, Marsha J and Colella, Phillip},
	Annote = {},
	Journal = {Journal of Computational Physics},
	Keywords = {gridding, adaptive mesh refinement},
	Number = {1},
	Pages = {64--84},
	Publisher = {Elsevier},
	Title = {Local adaptive mesh refinement for shock hydrodynamics},
	Volume = {82},
	Year = {1989}}



%  Berger 1984  (Gridding)
@article{berger1984adaptive,
	Author = {Berger, Marsha J and Oliger, Joseph},
	Annote = {},
	Keywords = {gridding, adaptive mesh refinement},
	Journal = {Journal of Computational Physics},
	Number = {3},
	Pages = {484--512},
	Publisher = {Elsevier},
	Title = {Adaptive mesh refinement for hyperbolic partial differential equations},
	Volume = {53},
	Year = {1984}}


% Bourgeat, 1984
@article{bourgeat1984homogenized,
	Author = {Bourgeat, Alain},
	Keywords = {two-phase flow, homogenization},
	Journal = {Computer Methods in Applied Mechanics and Engineering},
	Number = {1},
	Pages = {205--216},
	Publisher = {Elsevier},
	Title = {Homogenized behavior of two-phase flows in naturally fractured reservoirs with uniform fractures distribution},
	Volume = {47},
	Year = {1984}}




% Bramble and Hubbard - FDM
@article{bramble1962formulation,
  title={On the formulation of finite difference analogues of the Dirichlet problem for Poisson's equation},
  author={Bramble, James H and Hubbard, BE},
  journal={Numerische Mathematik},
  volume={4},
  number={1},
  pages={313--327},
  year={1962},
  publisher={Springer},
  apa={Bramble, J. H., \& Hubbard, B. E. (1962). On the formulation of finite difference analogues of the Dirichlet problem for Poisson's equation. Numerische Mathematik, 4(1), 313-327.}, 
  annote={}
}


% ------------------------------------ %
















% ----------------- C ------------------ %

% Caers 2005
@book{caers2005petroleum,
	Author = {Caers, Jef},
	Keywords = {Geostatistics},
	Publisher = {Society of Petroleum Engineers Richardson},
	Title = {Petroleum {G}eostatistics},
	Year = {2005}}



% Cardwell and Parsons
@article{cardwell1945average,
  title={Average permeabilities of heterogeneous oil sands},
  author={Cardwell Jr, WT and Parsons, RL and others},
  journal={Transactions of the AIME},
  volume={160},
  number={01},
  pages={34--42},
  year={1945},
  publisher={Society of Petroleum Engineers}, 
  annote={This paper discusses the practical problem of estimating a single equivalent permeability for an oil reservoir, or a portion thereof, whose actual permeability varies irregularly. Limiting averages for general types of permeability. variation is developed and illustrated by examples involving important, specific types of variation.  The theory of the flow of fluids through porous media is becoming increasingly important in predictions of oil-reservoir behavior. However, reservoirs are seldom found in practical applications to which simple theory strictly applies. Actual reservoirs have complicated shapes and non-uniform permeabilities and porosities. This paper discusses the problem of estimating a single equivalent permeability for an oil reservoir or a segment of an oil reservoir whose actual permeability varies irregularly. The equivalent permeability of a reservoir segment is defined as the permeability of a homogeneous segment of the same dimensions that would pass the same flux under the same pressure drop.The permeability value can be used in simple theoretical formulas to calculate the reservoir behavior.  To the authors' knowledge, previous theoretical calculations on systems of non-uniform permeability have been carried out only by Musk, and he has not dealt with irregular variations except in writing down the general differential equation for the pressure in variably permeable systems.  In practical work, many calculations have been made to estimate equivalent permeabilities from the permeability profiles obtained by core analyses. Regarding such calculations, Johnston and Sherborne say:  Simple arithmetic and weighted averages have been tried on many wells, and it has been found that, where frequent sampling has occurred, the arithmetic average is as satisfactory as a weighted average. The application of statistical methods to the analysis of permeability data, as recently presented by Laws, may prove fruitful. Law has made a valuable contribution in showing how statistical analyses may aid in picturing the characteristics of a reservoir from those of a necessarily limited number of core samples. It is believed, however, that the particular problem of the estimation of equivalent permeabilities can be most directly approached from a fluid dynamical viewpoint, as given in the present paper. An example of the use of the conclusions herein in conjunction with Law's method is presented.}, 
  keywords={Average permeabilities}
}






% Castellini
@inproceedings{castellini2000flow,
  Title={Flow based modules for grid generation in two and three dimensions},
  Author={Castellini, A and Edwards, Michael G and Durlofsky, Louis J},
  booktitle={ECMOR VII-7th European Conference on the Mathematics of Oil Recovery},
  Keywords = {grid generation, Durlofsky, Edwards},
  Annote={},
  Year={2000}
}



%Chen, Clauser, Marquart, ..
@article{chen2016modeling,
  title={Modeling anisotropic flow and heat transport by using mimetic finite differences},
  author={Chen, Tao and Clauser, Christoph and Marquart, Gabriele and Willbrand, Karen and B{\"u}sing, Henrik},
  journal={Advances in water resources},
  volume={94},
  pages={441--456},
  year={2016},
  publisher={Elsevier}
}


% Chen, Z., Huan, G., \& Ma, Y. (2006). Computational methods for multiphase flows in porous media (Vol. 2). Siam.
@book{chen2006computational,
  title={Computational methods for multiphase flows in porous media},
  author={Chen, Zhangxin and Huan, Guanren and Ma, Yuanle},
  volume={2},
  year={2006},
  publisher={Siam}, 
  annote={Textbook.  Covers motivation of reservoir simulation and numerical methods.  Chen, Z., Huan, G., \& Ma, Y. (2006). Computational methods for multiphase flows in porous media (Vol. 2). Siam.}
}


% chen 2015
@article{chen2015new,
	Author = {Chen, Tao and Clauser, Christoph and Marquart, Gabriele and Willbrand, Karen and Mottaghy, Darius},
	Journal = {Advances in water resources},
	Pages = {60--68},
	Publisher = {Elsevier},
	Title = {A new upscaling method for fractured porous media},
	Volume = {80},
	Year = {2015}, 
	Keywords = {Fractures media, permeability upscaling, flow-based upscaling, MPFA},
	Annote={Apply a recently developed multi-point flux approximation Finite Volume method for discrete fracture model simulation.}
}


% Chen
@article{chen2013time,
	Author = {Chen, Yuguang and Li, Yan and Efendiev, Yalchin},
	Journal = {Advances in water resources},
	Keywords = {two-phase flow, Subsurface flow, Time-of-flight, Upscaling Heterogeneity, relative permeability,Reservoir simulation},
	Pages = {119--132},
	Publisher = {Elsevier},
	Title = {Time-of-flight (TOF)-based two-phase upscaling for subsurface flow and transport},
	Volume = {54},
	Year = {2013}, 
	Annote = {Subsurface formations are characterized by heterogeneity over multiple length scales, which can strongly impact flow and transport. This paper presents a new upscaling approach based on time-of-flight (TOF) to generate upscaled two-phase flow functions. The method focuses on more accurate representations of local saturation boundary conditions, which have a dominant impact (compared to the pressure boundary conditions) on the upscaled two-phase flow models. The TOF-based upscaling approach effectively incorporates single-phase flow and transport information into local upscaling calculations, accounting for the global flow effects on saturation and the local variations due to subgrid heterogeneity. The method can be categorized into quasi-global upscaling techniques, as the global single-phase flow and transport information is incorporated in the local boundary conditions. Can the TOF-based two-phase upscaling be readily integrated into any existing local two-phase upscaling framework, thus making it more flexible than local-global two-phase upscaling approaches developed recently? The method was applied to permeability fields with different correlation lengths and fluid-mobility ratios. It was shown that the new method consistently outperforms existing local two-phase upscaling techniques, including recently developed methods with improved local boundary conditions (such as effective flux boundary conditions), and provides accurate coarse-scale models for both flow and transport.},
	apa = {Chen, Y., Li, Y., \& Efendiev, Y. (2013). Time-of-flight (TOF)-based two-phase upscaling for subsurface flow and transport. Advances in water resources, 54, 119-132.}
	}


% Chen
@book{chen2006computational,
  title={Computational methods for multiphase flows in porous media},
  author={Chen, Zhangxin and Huan, Guanren and Ma, Yuanle},
  volume={2},
  year={2006},
  publisher={SIAM}, 
  keywords={flow equations, multiphase flows}, 
  annote={} 
}




% chen 2006
@article{chen2006adaptive,
	Author = {Chen, Yuguang and Durlofsky, Louis J},
	Journal = {Transport in porous Media},
	Keywords = {Local-Global, tpfa},
	Number = {2},
	Pages = {157--185},
	Publisher = {Springer},
	Title = {Adaptive local--global upscaling for general flow scenarios in heterogeneous formations},
	Volume = {62},
	Year = {2006}, 
	Annote = {Uses an Local-Global strategy to compute transmissibilities based on one specific, well-driven global flow problem.  When boundary or well conditions change significantly, upscaled transmissibilities should be recomputed for this method.  This LG procedure used TPFA on Cartesian grids but could be applied to other grid topologies.  \hl{TPFA methods combined with local-global procedures provide good upscaled transmissibilities even in challenging cases}.  tpfa $+$ LG $=$ good}
	}


% chen 2003
@article{chen2003coupled,
Author = {Chen, Y. and Durlofsky, Louis J. and Gerritsen, M. and Wen, Xian-Huan-H.},
Title = {A coupled local--global upscaling approach for simulating flow in highly heterogeneous formations},
Keywords = {Local-global, Local-global = Quasi global method},
	Issn = {0309-1708},
	Journal = {Advances in Water Resources Advances in Water Resources},
	Number = {10},
	Pages = {1041-1060},
	Publisher = {Elsevier},
	Volume = {26},
	Year = {2003}, 
	Annote = { \textbf{Development of Local-global method}.   Quasi-global (Local-global) upscaling methods attempt to estimate the effects of global flow without solving a global fine-scale problem. These methods use global coarse-scale simulations to estimate the boundary conditions for the extended local calculation of  $T^*$. The procedure is iterated until the upscaled quantity is consistent with the global flow (i.e., self-consistency is enforced). Reduce computational requirements of full fine-scale global by using approximate global information instead of global fine-scale results).  Upscaled parameters are computed iteratively using local boundary conditions determined from generic global coarse-scale simulations.   Results also indicate that transmissibility upscaling is more accurate than permeability upscaling (when there is significant heterogeneity, i.e., channelized flows).  TPFA methods and local-global procedures provide good upscaled transmissibilities even in challenging cases.  In this work,  transmissibility upscaling is only considered using TPFA.  }}



% chen 2010
@article{chen2010global,
	Author = {Chen, Tianhong and Gerritsen, Margot G and Lambers, James V and Durlofsky, Louis J},
	Keywords = {VCMP},
	Journal = {Computational Geosciences},
	Number = {1},
	Pages = {65--81},
	Publisher = {Springer},
	Title = {Global variable compact multipoint methods for accurate upscaling with full-tensor effects},
	Volume = {14},
	Year = {2010}}

% chen 2008
@article{chen2008nonlinear,
	Author = {Chen, Yuguang and Mallison, Bradley T and Durlofsky, Louis J},
	Title = {Nonlinear two-point flux approximation for modeling full-tensor effects in subsurface flow simulations},
	Journal = {Computational Geosciences},
	Number = {3},
	Pages = {317--335},
	Publisher = {Springer},
	Volume = {12},
	Year = {2008},
	Keywords = {TPFA, Full-tensor effects},
	Annote = {Considers the open question of whether TPFA methods can be competitive with MPFA in terms of accuracy.  This paper partly addresses that question by applying global and local-global TPFA procedures to problems with significant full-tensor anisotropy. Subsurface flow models can exhibit strong full-tensor anisotropy due to either permeability or grid nonorthogonality effects. Upscaling procedures, for example, generate full-tensor effects on the coarse scale, even for cases where the underlying fine-scale permeability is isotropic. A multipoint flux approximation (MPFA) is often needed to simulate flow for such systems accurately. This paper presents and applies a different approach, nonlinear two-point flux approximation (NTPFA), for modeling systems with full-tensor effects. In NTPFA, transmissibility (which provides interblock connections) is determined from reference global flux and pressure fields for a specific flow problem. These fields can be generated using either fully resolved or approximate global simulations. Using fully resolved simulations leads to an NTPFA method corresponding to global upscaling procedures. In contrast, approximate simulations provide a method that corresponds to recently developed local-global techniques. NTPFA algorithms applicable to both single-scale full-tensor permeability systems and two-scale systems are described for both approaches. A unified framework is introduced, which enables single-scale and two-scale problems to be viewed consistently. Extensive numerical results demonstrate that global and local-global NTPFA techniques provide accurate flow predictions for single- and two-scale systems over wide parameter ranges. However, the global procedure is more accurate overall. The applicability of NTPFA to the simulation of two-phase flow in upscaled models is also demonstrated.}
	}

% chen
@article{chen2003mixed,
	Author = {Chen, Zhiming and Hou, Thomas},
	Journal = {Mathematics of Computation},
	Keywords = {Finite Element Upscaling methods, FEM},
	Number = {242},
	Pages = {541--576},
	Title = {A mixed multiscale finite element method for elliptic problems with oscillating coefficients},
	Volume = {72},
	Year = {2003}}



% Chen, QY
@article{chen2008enriched,
  title={Enriched multi-point flux approximation for general grids},
  author={Chen, Qian-Yong and Wan, Jing and Yang, Yahan and Mifflin, Rick T},
  journal={Journal of Computational Physics},
  volume={227},
  number={3},
  pages={1701--1721},
  year={2008},
  publisher={Elsevier}, 
  Annote ={},
  Keywords = {}
}



% Christie, 2001, 10th SPE Comparative Solution Project on Upscaling
@inproceedings{christie2001tenth,
	Annote = {Paper presents the results of the 10th SPE Comparative Solution Project on Upscaling. Two problems were chosen. The first problem was a small 2D gas-injection problem, chosen so that the fine grid could be computed easily and upscaling and pseudoization methods could be used.  The first problem was a simple, 2,000-cell 2D vertical cross-section. The specified tasks were to apply upscaling or pseudoization methods and to obtain solutions for a specified coarse grid and a coarse grid selected by the participant.},
	Author = {Christie, MA and Blunt, MJ and others},
	Keywords = {Reservoir Simulation },
	Booktitle = {SPE Reservoir Simulation Symposium},
	Organization = {Society of Petroleum Engineers},
	Title = {Tenth {SPE} comparative solution project: A comparison of upscaling techniques},
	Year = {2001}}

% Christie

@article{christie1996upscale,
	Author = {Christie,M. A.},
	Date-Modified = {2020-05-02 13:24:29 -0400},
	Journal = {Journal of Petroleum Technology},
	Keywords = {upscaling, Survey of upscaling, limitations, correlation length, large-aspect-ratio gridblocks},
	Number = {11},
	Pages = {1004-1010},
	Title = {Upscaling for Reservoir Simulation},
	Volume = {48},
	Year = {1996},
	Abstract = {Summary: Upscaling has become an increasingly important tool in recent years for converting highly detailed geological models to simulation grids. This paper reviews and summarizes both single- and two-phase upscaling techniques.  

Introduction: A principal motivation for developing upscaling techniques has been the development of geostatistical reservoir description algorithms.1-3 These algorithms now routinely result in fine-scale descriptions of reservoir porosity and permeability on grids of tens of millions of cells. The descriptions honor the known and inferred statistics of the reservoir properties. Fig. 1 shows an example of such a reservoir description. These reservoir-description grids are far too fine to be used as grids in reservoir simulators. Despite advances in computer hardware, most full-field reservoir models still use fewer than 100,000 cells, a factor of 100 down on the geological grid.  Upscaling is needed to bridge the gap between these two scales. Given a fine-scale reservoir description and a simulation grid, an upscaling algorithm assigns suitable values for porosity, permeability, and other flow functions to cells on the coarse simulation grid. Many possible choices for an upscaling approach exist; see Refs. [4] through [7] for examples.  

Single-Phase Upscaling:  The simplest form of upscaling is single-phase upscaling. Here, the aim is to preserve the gross flow features on the simulation grid. 

The algorithm calculates an "effective permeability," which results in the same total flow of single-phase fluid through the coarse, homogeneous block as that obtained from the fine heterogeneous block.  

Pressure-Solver Methods:  In the pressure-solver method, we set up a single-phase-flow calculation with specified boundary conditions and then ask what value of effective permeability yields the same flow rate as the fine-grid calculation.  The results we obtain depend on our assumptions, particularly about boundary conditions. The most common assumption is that no-flow boundary conditions exist on the cube's walls. This gives rise to a diagonal tensor that can be entered directly into a reservoir simulator.  Although almost all the fine details have been lost, the broad features are retained.  

Directional Effective Permeabilities: To calculate directional effective permeabilities, we set up calculations in the x, y, and z directions as follows.  Set up a matrix equation to solve with no-flow boundary conditions along the sides, p=1 at the inlet, and p=0 at the outlet. Solve the equation and sum the fluxes in the x direction.  This approach is simple and very effective in most circumstances. For example, Begget al.8 obtained effective vertical permeabilities using pressure-solver techniques that agreed very closely with the values received with a history-matching technique.

Full-Tensor Effective Permeabilities:  Alternatively, some authors [9-13] assume periodic boundary conditions and calculate a full-tensor effective permeability.  This is significantly more accurate but has the disadvantage that it cannot be directly entered into a commercial reservoir simulator. Tensor-effective permeabilities are still the subject of active research, particularly in symmetry.  Durlofsky [14] gives a good summary of scale-up involving tensor permeabilities.  He favors the application of periodic boundary conditions.   His approach is supported by Pickup [13], who compared the accuracy and robustness of several boundary conditions in calculating effective permeabilities.  

Renormalization Techniques: Renormalization methods offer a faster but less accurate method of calculating an effective permeability. For most cases, renormalization gives effective permeabilities close to the direct solution of the pressure equation and allows rapid calculation of effective permeabilities from extensive grid systems. The renormalization approach works by taking a significant problem and breaking it down into a hierarchy of manageable problems. It has proven successful in a variety of theoretical physics areas.The renormalization method for effective permeabilities was pioneered by King [15], who used a resister-network analogy to write down direct expressions for effective permeabilities on sequences of 2x2 cells. Fig. 4 shows the procedure.  A small group of cells is extracted, then the effective permeability is calculated and put back in place of the original fine group of cells. This can be repeated for many levels and gives a fast estimation of effective permeability. Renormalization is not limited to 2x2 cells and resister-network analogies and can be coded for arbitrary scale changes between levels through direct methods for matrix inversion. 

Other Techniques: Other techniques that should be mentioned include effective medium theory [15], power-law averaging [16], harmonic-/arithmetic-mean techniques, and homogenization theories [17].  These techniques are generally very fast but suffer from some limitations in applicability.  

Limitations in Upscaling:  One of the main limitations of upscaling is that it usually answers with almost no indication of whether the assumptions made in deriving the answer hold. Limited attempts have been made to analyze the upscaling process [18], but no good theory exists that unequivocally states whether an upscaled value provides a good or bad approximation.  Some areas are known to raise concern about whether the upscaled values are good approximations; these include large-aspect-ratio grid blocks, significant transport at an angle to the grid lines, and upscaled grid blocks close in size to a correlation length of the system. The main practical advice that can be given under these circumstances is to try to vary the parameters that cause concern. For example, where correlation lengths are close to upscaled gridblock sizes, you can see a significant change in upscaled value for coarse-grid sizes of half or twice the original size.  Another factor is that the effective permeability values depend on the difference operator used to solve the pressure equations and the permeabilities on the underlying fine grid. This can be particularly important for large-aspect-ratio grid blocks typically used to calculate effective vertical permeabilities. Some recent work by Edwards [19] offers the potential in reducing the effects of this gridlock aspect-ratio problem.},
	Annote = {Geostatistical reservoir description algorithms routinely result in fine-scale descriptions of reservoir porosity and permeability on grids of tens of millions of cells.  Reservoir-description grids are far too fine to be used as grids in reservoir simulators. Given a fine-scale reservoir description and a simulation grid, an upscaling algorithm assigns suitable values for porosity, permeability, and other flow functions to cells on the coarse simulation grid. Many possible choices of upscaling approach exist.  The algorithm calculates model parameters that result in the same total flow of single-phase fluid through the coarse, homogeneous block as that obtained from the fine heterogeneous block (i.e., Pressure-Solver Methods = set up a single-phase-flow calculation with specified boundary conditions and then asks what value of transmissibility (or permeability) yields the same flow rate as the fine-grid calculation.  Fig. 3 shows an example of a two-dimensional calculation that scaled a 128 x 128 fine grid up to an 8 x 8 coarse grid. Although almost all the fine details have been lost, the broad features are retained.  Other techniques are generally very fast but suffer from some limitations in applicability (including renormalization, power-law averaging, etc.).  Some areas are known to raise concern about whether the up-scaled values are good approximations; these include large-aspect-ratio grid blocks, significant transport at an angle to the grid lines, and upscaled grid blocks close in size to a correlation length of the system.   Effective permeability values depend on the difference operator used to solve the pressure equations and the permeabilities on the underlying fine grid.  It is necessary to be aware of some key areas when performing any form of upscaling. The size and shape of the gridblocks used will affect the answer obtained.}
	}



% ------------------------------------ %
















% ----------------- D ------------------ %

% Darcy
@article{darcy1856fontaines,
  title={Les fontaines publiques de la ville de Dijon, Dalmont},
  author={Darcy, Henry},
  journal={Paris: Dalmont},
  year={1856}, 
  Keywords = {Darcy's Law},
  Annote={}
}


% Davis:  AMR
@incollection{davis2017adaptive,
  title={Adaptive Mesh Refinement: An Essential Ingredient in Computational Science.},
  author={Davis, P},
  booktitle={SIAM News},
  year={2017}, 
  Keywords = {AMR, Gridding},
  Annote={}
}

% Deutsch
@article{deutsch1998geostatistical,
	Author = {Deutsch, Clayton V and Journel, Andre G},
	Journal = {Oxford University Press, New York},
	Title = {Geostatistical software library and user's guide},
	Year = {1998},
	Keywords = { }
	}




% Dickson - groundwater resource management
@article{dickson2014coupling,
  title={Coupling ground and airborne geophysical data with upscaling techniques for regional groundwater modeling of heterogeneous aquifers: Case study of a sedimentary aquifer intruded by volcanic dykes in {N}orthern {I}reland},
  author={Dickson, Neil Edwin Matthew and Comte, Jean-Christophe and McKinley, Jennifer and Ofterdinger, Ulrich},
  journal={Water Resources Research},
  volume={50},
  number={10},
  pages={7984--8001},
  year={2014},
  publisher={Wiley Online Library}
}




% Durlofsky
@inproceedings{durlofsky2005upscaling,
	Author = {Durlofsky, Louis J},
	Booktitle = {8th International Forum on Reservoir Simulation Iles Borromees, Stresa, Italy},
	Keywords = {Survey of upscaling, gridding},
	Title = {Upscaling and gridding of fine scale geological models for flow simulation},
	Volume = {2024},
	Year = {2005},
		Annote = {Survey of upscaling.  Review of a variety of approaches for gridding and upscaling of detailed geocellular models for flow simulation.  Coarse scale simulation results generated using many of these upscaling techniques are presented. These results illustrate the capabilities of existing upscaling procedures and demonstrate the levels of accuracy attainable using the various approaches.}
		}


% Durlofsky
@article{durlofsky2007adaptive,
	Author = {Durlofsky, LJ and Efendiev, Y and Ginting, V},
	Journal = {Advances in Water Resources},
	Keywords = {Local-Global},
	Number = {3},
	Pages = {576--588},
	Publisher = {Elsevier},
	Title = {An adaptive local--global multiscale finite volume element method for two-phase flow simulations},
	Volume = {30},
	Year = {2007}}


% Durlofsky
@article{durlofsky1991numerical,
	Author = {Durlofsky, Louis J.},
	Keywords = {FEM, Flow-based modelling, Permeability },
	Journal = {Water resources research},
	Number = {5},
	Pages = {699-708},
	Publicationstatus = {Published},
	Publisher = {Wiley Online Library},
	Title = {Numerical calculation of equivalent grid block permeability tensors for heterogeneous porous media},
	Volume = {27},
	Year = {1991}, 
    Annote = {A numerical procedure for determining equivalent gridblock permeability tensors for heterogeneous porous media is presented.  Triangle-based Finite Element procedure solves pressure equations on the fine scale.  The specification of periodic boundary conditions is shown to yield symmetric, positive, definite equivalent permeability tensors in all cases.  The applicability and limitations of the method for more general heterogeneity fields are discussed.}}


% Durlofsky
@article{durlofsky1992representation,
	Author = {Durlofsky, Louis J},
	Keywords = { },
	Journal = {Water Resources Research},
	Number = {7},
	Pages = {1791--1800},
	Publisher = {Wiley Online Library},
	Title = {Representation of grid block permeability in coarse scale models of randomly heterogeneous porous media},
	Volume = {28},
	Year = {1992},
    Annote = {By comparing the flow results in the grid at the measurement (fine) scale and the flow results in the upscaled grid, they concluded that no simple average is valid for all heterogeneous formations. }
}


	
% Durlofsky
@article{durlofsky1993triangle,
  title={A triangle based mixed finite element-finite volume technique for modeling two phase flow through porous media},
  author={Durlofsky, Louis J},
  journal={Journal of Computational Physics},
  volume={105},
  number={2},
  pages={252--266},
  year={1993},
  publisher={Elsevier}, 
  annote={}, 
  keywords={}
}



% ------------------------------------ %
















% ----------------- E ------------------ %

% Edwards
@article{edwards1996elimination,
	Author = {Edwards, Michael G},
	Keywords = { },
	Journal = {Journal of Computational Physics},
	Number = {2},
	Pages = {356--372},
	Publisher = {Elsevier},
	Title = {Elimination of adaptive grid interface errors in the discrete cell centered pressure equation},
	Volume = {126},
	Year = {1996}}


% Edwards
@article{edwards1998finite,
	Author = {Edwards, Michael G and Rogers, Clive F},
	Keywords = {Multipoint discretization, MPFA},
	Annote={Present a new family of flux continuous, locally conservative, finite volume schemes applicable to the diagonal and full tensor pressure equations with generally discontinuous coefficients.},
	Journal = {Computational Geosciences},
	Number = {4},
	Pages = {259--290},
	Publisher = {Springer},
	Title = {Finite volume discretization with imposed flux continuity for the general tensor pressure equation},
	Volume = {2},
	Year = {1998}}


% Edwards
@article{edwards2002unstructured,
  title={Unstructured, control-volume distributed, full-tensor finite-volume schemes with flow based grids},
  author={Edwards, Michael G},
  journal={Computational Geosciences},
  volume={6},
  number={3-4},
  pages={433--452},
  year={2002},
  publisher={Springer}, 
  Annote={Locally conservative flux-continuous, full-tensor discretization schemes are presented for general unstructured grids. The schemes are control-volume distributed, where flow variables and rock properties are assigned to the polygonal control-volumes derived from the primal grid. A relationship between these finite volume schemes and the mixed finite element method is established. An extension for unstructured grids is described that leads to a general symmetric positive definite discretization matrix for both quadrilateral and triangular grids. A novel flow-based gridding approach for unstructured mesh generation is also proposed for heterogeneous reservoir domains. Results computed with the flux continuous schemes on unstructured flow-based grids demonstrate the advantages of the methods.}
}




% Edwards
@article{edwards2000m,
  title={M-matrix flux splitting for general full tensor discretization operators on structured and unstructured grids},
  author={Edwards, Michael G},
  journal={Journal of Computational Physics},
  volume={160},
  number={1},
  pages={1--28},
  year={2000},
  publisher={Elsevier}, 
  Keywords = {unstructured grids, Full-tensor effects},
  annote={}
}



% Efendiev
@misc{efendiev2009multiscale,
	Author = {Efendiev, Yalchin and Hou, Thomas Y},
	Keywords = {Multiscale, FEM},
	Publisher = {Springer, New York},
	Title = {Multiscale finite element methods, volume 4 of Surveys and Tutorials in the Applied Mathematical Sciences},
	Year = {2009}}
	
	
	
% Eigestad	
@article{eigestad2002symmetry,
  title={Symmetry and M-matrix issues for the {O}-method on an unstructured grid},
  author={Eigestad, GT and Aavatsmark, I and Espedal, M},
  journal={Computational Geosciences},
  volume={6},
  number={3-4},
  pages={381--404},
  year={2002},
  publisher={Springer}, 
  keywords={MPFA, unstructured grids, MPFA-O}, 
  annote={Focuses on mathematical properties of the discrete operator}
}


% Eigestad
@article{eigestad2005convergence,
  title={On the convergence of the multi-point flux approximation {O}-method: {N}umerical experiments for discontinuous permeability},
  author={Eigestad, GT and Klausen, RA},
  journal={Numerical Methods for Partial Differential Equations: An International Journal},
  volume={21},
  number={6},
  pages={1079--1098},
  year={2005},
  publisher={Wiley Online Library},
    annote={}, 
  abstract={}, 
  apa={Eigestad, G. T., \& Klausen, R. A. (2005). On the convergence of the multi?point flux approximation O?method: Numerical experiments for discontinuous permeability. Numerical Methods for Partial Differential Equations: An International Journal, 21(6), 1079-1098.
}, 
  keywords={MPFA, convergence, O-method}, 
  ams={35, 76}
}



% Eymard
@article{eymard1999convergence,
  title={Convergence of finite volume schemes for semilinear convection diffusion equations},
  author={Eymard, Robert and Gallou{\"e}t, Thierry and Herbin, Rapha{\`e}le},
  journal={Numerische Mathematik},
  volume={82},
  number={1},
  pages={91--116},
  year={1999},
  publisher={Springer}, 
  annote={}, 
  abstract={}, 
  apa={}, 
  keywords={convergence, FVM}, 
  ams={}
}


% Eymard
@article{eymard2001finite,
  title={Finite volume approximation of elliptic problems and convergence of an approximate gradient},
  author={Eymard, Robert and Gallou{\"e}t, Thierry and Herbin, Raphaele},
  journal={Applied Numerical Mathematics},
  volume={37},
  number={1-2},
  pages={31--53},
  year={2001},
  publisher={Elsevier}, 
  annote={}, 
  abstract={}, 
  apa={Eymard, R., Gallou{\"e}t, T. and Herbin, R., 2001. Finite volume approximation of elliptic problems and convergence of an approximate gradient. Applied Numerical Mathematics, 37(1-2), pp.31-53.}, 
  keywords={}, 
  ams={}
}



% ------------------------------------ %
















% ----------------- F ------------------ %

% Farmer
@article{farmer2002upscaling,
	Annote = {Definition of Upscaling, review of two-stage upscaling methods reviewed (e.g., local-global)...first word refers to the experiment and the second to the method of calibration},
	Author = {Farmer, C. L.},
	Journal = {International journal for numerical methods in fluids},
	Keywords = {Upscaling, Fine-scale, coarse-scale calibration},
	Number = {1-2},
	Pages = {63-78},
	Publicationstatus = {Published},
	Publisher = {Wiley Online Library},
	Title = {Upscaling: a review},
	Volume = {40},
	Year = {2002}}






@article{friis2008,
author={Friis,Helmer A. and Edwards,Michael G. and Mykkeltveit,Johannes},
year={2008},
title={Symmetric Positive Definite Flux-Continuous Full-Tensor Finite-Volume Schemes on Unstructured Cell-Centered Triangular Grids},
journal={SIAM Journal on Scientific Computing},
volume={31},
number={2},
pages={1192-29},
note={Copyright - Copyright] © 2008 Society for Industrial and Applied Mathematics; Last updated - 2012-06-29},
abstract={Novel cell-centered full-tensor finite-volume methods are presented for general unstructured grids in two spatial dimensions. The numerical schemes are flux-continuous and based on computing the transmissibilities in a local subcell transform space, ensuring that local flux matrices are symmetric. As a result the global discretization matrix is shown to be symmetric positive definite for any grid type. A symmetric physical space method is also introduced, and the symmetric methods are shown to be closely related. Discrete ellipticity conditions are derived for positive definiteness of the physical space and subcell space schemes. Computational examples are presented for unstructured triangular grids demonstrating good performance of the scheme. The schemes are compared with the so-called multipoint flux approximation (MPFA) O-method I. Aavatsmark, T. Barkve, Ø. Bøe, and T. Mannseth, SIAM J. Sci. Comput., 19 (1998), pp. 1700-1716]. Good agreement between the methods is obtained, but the new scheme shows improved behavior in challenging cases.},
keywords={Mathematics},
annote={},
apa={Friis, H. A., Edwards, M. G., \& Mykkeltveit, J. (2008). Symmetric positive definite flux-continuous full-tensor finite-volume schemes on unstructured cell-centered triangular grids. SIAM Journal on Scientific Computing, 31(2), 1192-29. doi:http://dx.doi.org.une.idm.oclc.org/10.1137/070692182}
}

% ------------------------------------ %
















% ----------------- G ------------------ %

% Gao and Wu
@article{gao2015second,
  title={A second-order positivity-preserving finite volume scheme for diffusion equations on general meshes},
  author={Gao, Zhiming and Wu, Jiming},
  journal={SIAM Journal on Scientific Computing},
  volume={37},
  number={1},
  pages={A420--A438},
  year={2015},
  publisher={SIAM}
}



% Gautier, Blunt,\& Christie 1999
@article{gautier1999nested,
	Author = {Gautier, Yann and Blunt, Martin J and Christie, Michael A},
	Keywords = { },
	Journal = {Computational Geosciences},
	Number = {3},
	Pages = {295--320},
	Publisher = {Springer},
	Title = {Nested gridding and streamline-based simulation for fast reservoir performance prediction},
	Volume = {3},
	Year = {1999}}



% Gerritsen\& Durlofsky 2005
@article{gerritsen2005modeling,
	Annote={ }, 
	Author = {Gerritsen, Margot G and Durlofsky, Louis J},
	Journal = {Annu. Rev. Fluid Mech.},
	Keywords = {Multiscale methods},
	Pages = {211--238},
	Publisher = {Annual Reviews},
	Title = {Modeling fluid flow in oil reservoirs},
	Volume = {37},
	Year = {2005}}





% Gerritsen\& Lambers, 2008
@article{gerritsen2008integration,
	Annote = {Introduces MLLG method with (Multi-Level = grid adaptivity), where as Chen (2003) introduced only LG).  MLLG improves representation of large scale connectivities and reduces process dependency(*definition?).  Grid adaptivity reduces upscaling errors and improves accuracy in high-flow path resolution.  After the global solve, coarse face fluxes are calculated and flagged for refinement if flux exceeds a certain threshold.  },
	Author = {Gerritsen, M. and Lambers, J.V.},
	Journal = {Computational Geosciences},
	Keywords = {Local-global, grid adaptivity},
	Number = {2},
	Pages = {193--208},
	Publisher = {Springer},
	Title = {Integration of local--global upscaling and grid adaptivity for simulation of subsurface flow in heterogeneous formations},
	Volume = {12},
	Year = {2008}}



% Gerritsen 2007
@article{gerritsen2007solving,
	Author = {Gerritsen, MG and Lambers, JV},
	Journal = {To be submitted to Journal of Computational Physics},
	Keywords = {3D, anisotropic adaptation, adaptive grid, gridding, elliptic equation},
	Title = {Solving elliptic equations in heterogeneous media using Cartesian grid methods with anisotropic adaptation},
	Year = {2007}}



% Gerritsen 2006
@inproceedings{gerritsen2006variable,
	Author = {Gerritsen, MG and Lambers, JV and Mallison, BT},
	Keywords = {VCMP, Monotone, MPFA, transmissibility upscaling},
	Annote = { },
	Booktitle = {Proceedings of the 10th European Conference on the Mathematics of Oil Recovery, Amsterdam},
	Title = {A variable and compact MPFA for transmissibility upscaling with guaranteed monotonicity},
	Volume = {47},
	Year = {2006}}



% Gerritsen 2005
@inproceedings{gerritsen2005fully,
	Annote = {Conference presenting CCAR framework(?)},
	Author = {Gerritsen, Margot Geertrui and Jessen, Kristian and Mallison, Bradley T and Lambers, James Vincent and others},
	Keywords = { },
	Booktitle = {SPE Annual Technical Conference and Exhibition},
	Organization = {Society of Petroleum Engineers},
	Title = {A fully adaptive streamline framework for the challenging simulation of gas-injection processes},
	Year = {2005}}




%  Gill, 1981
@article{gill1981practical,
	Author = {Gill, Philip E and Murray, Walter and Wright, Margaret H},
	Keywords = {optimization},
	Annote = { },
	Publisher = {Academic press},
	Title = {Practical optimization},
	Year = {1981}}



% Golub
@article{golub1999tikhonov,
  title={Tikhonov regularization and total least squares},
  author={Golub, Gene H and Hansen, Per Christian and O'Leary, Dianne P},
  journal={SIAM {J}ournal on {M}atrix {A}nalysis and {A}pplications},
  volume={21},
  number={1},
  pages={185--194},
  year={1999},
  publisher={SIAM}
}

% Gomez, Journel 1994
@article{gomez1994stochastic,
	Author = {Gomez-Hernandez, J Jaime and Journel, Andre},
	Keywords = { },
	Annote = { },
	Journal = {SPE Formation Evaluation},
	Number = {02},
	Pages = {93--99},
	Publisher = {Society of Petroleum Engineers},
	Title = {Stochastic characterization of gridblock permeabilities},
	Volume = {9},
	Year = {1994}}



@article{guerillot2019transmissibility,
  title={Transmissibility Upscaling on Unstructured Grids for Highly Heterogeneous Reservoirs},
  author={Gu{\'e}rillot, Dominique and Bruyelle, J{\'e}r{\'e}mie},
  journal={Water},
  volume={11},
  number={12},
  pages={2647},
  year={2019},
  publisher={Multidisciplinary Digital Publishing Institute}, 
  keywords={upscaling; transmissibility; heterogeneous reservoirs; reservoir simulations; unstructured grid; multi-scale; hydrogeology; reservoir engineering; numerical method; homogenisation; flow in porous media; compositional modelling; multi-phase flow; rock-fluid interactions; geological model}, 
  apa={Gu{\'e}rillot, D \& Bruyelle, J{\'e}r{\'e}mie (2019). Transmissibility Upscaling on Unstructured Grids for Highly Heterogeneous Reservoirs. Water, 11(12), 2647.}, 
  annote={}, 
  Abstract={One critical point of modeling flow in porous media is the capacity to consider highly variable parameters in space. It is then very challenging to simulate numerical fluid flow on such heterogeneous porous media. The continuous increase in computing power makes it possible to integrate smaller and smaller heterogeneities into geological models of up to tens of millions of cells. On such meshes, despite computer performance, multi-phase flow equations cannot be solved in an acceptable time for hydrogeologists and reservoir engineers, especially when the modeling considers several components in each fluid and when rock-fluid interactions are considered. Taking average reservoir properties is a common approach to reducing mesh size. During the last decades, many authors have studied the upscaling topic. Two ways have been investigated to upscale the absolute permeability: (1) an average of the permeability for each cell, which is then used for standard transmissibility calculation, or (2) directly computing upscaled transmissibility values using the high-resolution permeability values. This paper is related to the second approach. The proposed method uses the half-block approach and combines the finite volume principles with algebraic methods to provide an upper and a lower bound of the upscaled transmissibility values. An application on an extracted map of the SPE10 model shows that this approach is more accurate and faster than the classical transmissibility upscaling method based on flow simulation. This approach keeps the contrast of transmissibility values observed at the high-resolution geological scale and improves the accuracy of field-scale flow simulation for highly heterogeneous reservoirs. Moreover, the upper and lower bounds delivered by the algebraic method allow checking the quality of the upscaling and the gridding.}
}




% ------------------------------------ %
















% ----------------- H ------------------ %

% Ham 2002
@article{ham2002cartesian,
	Author = {Ham, FE and Lien, FS and Strong, AB},
	Keywords = {Cartesian grid , anisotropic adaptation, adaptive refinement},
	Annote = { },
	Journal = {Journal of Computational Physics},
	Number = {2},
	Pages = {469--494},
	Publisher = {Elsevier},
	Title = {A Cartesian grid method with transient anisotropic adaptation},
	Volume = {179},
	Year = {2002}}



% He, 2006
@article{he2006structured,
	Author = {He, C and Durlofsky, LJ},
	Keywords = {Upscaling, gridding},
	Annote = { },
	Journal = {Advances in water resources},
	Number = {12},
	Pages = {1876--1892},
	Publisher = {Elsevier},
	Title = {Structured flow-based gridding and upscaling for modeling subsurface flow},
	Volume = {29},
	Year = {2006}}


% He, et al
@article{he2002numerical,
  title={Numerical calculation of equivalent cell permeability tensors for general quadrilateral control volumes},
  author={He, Chuanping and Edwards, Michael G and Durlofsky, Louis J},
  journal={Computational Geosciences},
  volume={6},
  number={1},
  pages={29--47},
  year={2002},
  publisher={Springer},
  annote={A new method for upscaling fine scale permeability fields to general quadrilateral-shaped coarse cells is presented. The procedure, referred to as the conforming scale up method, applies a triangle-based finite element technique, capable of accurately resolving both the coarse cell geometry and the subgrid heterogeneity, to the solution of the local fine scale problem. An appropriate averaging of this solution provides the equivalent permeability tensor for the coarse scale quadrilateral cell. The general level of accuracy of the technique is demonstrated through application to a number of flow problems. The real strength of the conforming scale up method is demonstrated when the method is applied in conjunction with a flow-based gridding technique. In this case, the approach is shown to provide results that are significantly more accurate than those obtained using standard techniques.}
}



% Herbin 2008
@inproceedings{herbin2008benchmark,
	Annote = {Linear schemes (i.e., MPFA), are second-order accurate on unstructured meshes, but do not alway preserve the non-negativity of the solution for distored meshes or with high anisotropy ratio.},
	Author = {Herbin, Rapha{\`e}le and Hubert, Florence},
	Booktitle = {Finite volumes for complex applications V},
	Keywords = {monotone, non-negative pressure solution, anisotropic},
	Organization = {Wiley},
	Pages = {659--692},
	Title = {Benchmark on discretization schemes for anisotropic diffusion problems on general grids},
	Year = {2008}}



% Hesse 2008
@article{hesse2008compact,
	Author = {Hesse, Marc A and Mallison, Bradley T and Tchelepi, Hamdi A},
	Keywords = {multiscale finite volume, MSFV, elliptic, anisotropic},
	Annote = {For heterogeneous cases the compact 7-pt stencil improves the monotonicity of the MSFV method },
	Journal = {Multiscale Modeling \& Simulation},
	Number = {2},
	Pages = {934--962},
	Publisher = {SIAM},
	Title = {Compact multiscale finite volume method for heterogeneous anisotropic elliptic equations},
	Volume = {7},
	Year = {2008}}



% Holden 2000
@article{holden2000global,
	Title = {Global upscaling of permeability in heterogeneous reservoirs: The Output Least Squares (OLS) Method},
	Author = {Holden, Lars and Nielsen, Bj{\o}rn Fredrik},
	Keywords = {Global Upscaling Permeability},
	Annote = {Presents a new technique for computing the effective permeability on a coarse scale.  Traditional upscaling methods depend on local boundary conditions. It is well known that the permeability may depend heavily on the local boundary condition chosen. Hence the estimate is not stable. We propose to compute a coarse scale permeability field that minimizes the error, measured in a global norm, in the velocity and pressure fields. This leads to stable problems for a large number of reservoirs. We present several algorithms for finding the effective permeability values. It turns out that these algorithms are not significantly more computational expensive than traditional local methods},
	Journal = {Transport in Porous Media},
	Number = {2},
	Pages = {115--143},
	Publisher = {Springer},
	Volume = {40},
	Year = {2000}}




% Holden 1992
@article{holden1992tensor,
	Annote = {In an oil reservoir, the reservoir performance is influenced by heterogeneities on all scales. The size of the fluctuation in absolute permeability can be severe, ranging over many orders of magnitude. This makes it difficult to assign to each block a single effective value which gives the same mean flow. Many attempts have been made to address this problem. Several methods proposed earlier have either been limited to certain classes of heterogeneity distributions or to the calculation of diagonal permeability tensors. This paper goes a step further by proposing a method for calculating a fully effective permeability tensor using an iterative method.},
	Author = {Holden, Lars and Lia, Oddvar},
	Keywords = {Perm Upscaling, calculating effective permeability},
	Journal = {Transport in Porous Media},
	Number = {1},
	Pages = {37--46},
	Publisher = {Springer},
	Title = {A tensor estimator for the homogenization of absolute permeability},
	Volume = {8},
	Year = {1992}}




% Holstein, SPE 
@article{holstein2007volume,
  title={Volume V--Reservoir Engineering and Petrophysics},
  author={Holstein, Edward D},
  journal={Petroleum Engineering Handbook; Lake, LW, Ed.; Society of Petroleum Engineers: Richardson, TX},
  year={2007}, 
  keywords={}, 
  apa={Holstein, E. D. (2007). Volume V?Reservoir Engineering and Petrophysics. \textit{Petroleum Engineering Handbook; Lake, LW, Ed.; Society of Petroleum Engineers}: Richardson, TX.
},
  annote={},
  abstract={}
}



% Hornung, 1997
@article{hornung1997adaptive,
	Author = {Hornung, Richard D and Trangenstein, John A},
	Keywords = {},
	Annote = {Numerical simulation of these flows can be challenging and computationally expensive. Dynamic adaptive mesh optimisation and related approaches, such as adaptive grid refinement can increase solution accuracy at reduced computational cost.},
	Journal = {Journal of computational Physics},
	Number = {2},
	Pages = {522--545},
	Publisher = {Elsevier},
	Title = {Adaptive mesh refinement and multilevel iteration for flow in porous media},
	Volume = {136},
	Year = {1997}}





% Hou
@article{hou1997multiscale,
	Author = {Hou, Thomas Y and Wu, Xiao-Hui},
	Keywords = {Finite Element Upscaling Method, FEM},
	Journal = {Journal of computational physics},
	Number = {1},
	Pages = {169--189},
	Publisher = {Elsevier},
	Title = {A multiscale finite element method for elliptic problems in composite materials and porous media},
	Volume = {134},
	Year = {1997}}












% ----------------- I ------------------ %

% Islam
@book{islam2016advanced,
  title={Advanced Petroleum Reservoir Simulation: Towards Developing Reservoir Emulators},
  author={Islam, M Rafuqul and Hossain, M Enamul and Mousavizadegan, S Hossien and Mustafiz, Shabbir and Abou-Kassem, Jamal H},
  year={2016},
  publisher={John Wiley \& Sons}, 
  keywords={Book, Petroleum Reservoir Simulation}, 
  annote={}, 
  ams={}, 
  apa={Islam, M. R., Hossain, M. E., Mousavizadegan, S. H., Mustafiz, S., \& Abou-Kassem, J. H. (2016). Advanced Petroleum Reservoir Simulation: Towards Developing Reservoir Emulators. John Wiley \& Sons.}
}






% ----------------- J ------------------ %


% Jamal
@misc{jamal2006petroleum,
  title={Petroleum reservoir simulation: a Basic Approach},
  author={Jamal, H and SM, Farouq Ali and M Rafiq, Islam and others},
  year={2006},
  publisher={Gulf Publishing Company}, 
  keywords={book}, 
  annote={}, 
  ams={}, 
  apa={Jamal, H., SM, F. A., \& M Rafiq, I. (2006). Petroleum reservoir simulation: a Basic Approach.}
}



% Jenny 
@article{jenny2003multi,
	Annote = {\hl{Multi-scale Transmissibility upscaling using MPFA}. Chen, Durlofsky, Gerritsen, Wen paper cites this paper as performing transmissibility upscaling using MPFA (instead of only TPFA like they did in their paper). In this paper we present a multi-scale finite-volume (MSFV) method to solve elliptic problems with many spatial scales arising from flow in porous media. The method efficiently captures the effects of small scales on a coarse grid, is conservative, and treats tensor permeabilities correctly. The underlying idea is to construct transmissibilities that capture the local properties of the differential operator. This leads to a multi-point discretization scheme for the finite- volume solution algorithm. Transmissibilities for the MSFV have to be constructed only once as a preprocessing step and can be computed locally. Therefore this step is perfectly suited for massively parallel computers. Furthermore, a conservative fine-scale velocity field can be constructed from the coarse-scale pressure solution. Two sets of locally computed basis functions are employed. The first set of basis functions captures the small-scale heterogeneity of the underlying permeability field, and it is computed in order to construct the effective coarse-scale transmissibilities. A second set of basis functions is required to construct a conservative fine-scale velocity field. The accuracy and efficiency of our method is demonstrated by various numerical experiments. },
	Author = {Jenny, Patrick and Lee, SH and Tchelepi, Hamdi A},
	Keywords = {transmissibility upscaling using MPFA},
	Journal = {Journal of Computational Physics},
	Number = {1},
	Pages = {47--67},
	Publisher = {Elsevier},
	Title = {Multi-scale finite-volume method for elliptic problems in subsurface flow simulation},
	Volume = {187},
	Year = {2003}}



% Journel, 1986
@article{journel1986geostatistics,
	Author = {Journel, AG},
	Keywords = {geostatistics, isotropic, permeability},
	Annote = {Fine-scale permeability field is commonly isotropic},
	Journal = {Mathematical Geology},
	Number = {1},
	Pages = {119--140},
	Publisher = {Springer},
	Title = {Geostatistics: models and tools for the earth sciences},
	Volume = {18},
	Year = {1986}}


@inproceedings{journel1986power,
  title={Power averaging for block effective permeability},
  author={Journel, AG and Deutsch, C and Desbarats, AJ and others},
  booktitle={SPE California Regional Meeting},
  year={1986},
  organization={Society of Petroleum Engineers}, 
  annote={}, 
  keywords = {}, 
  abstract = {A binary type permeability distribution with spatial autocorrelation is introduced to model the transition between shales and sand in a reservoir. The contrast between the two modal permeability values can be made realistically high, and the autocorrelation ranges can be made realistically large and anisotropic. A steady state, single phase flow simulation is run over a network whose block grids are informed from the previous permeability distribution. The resulting network effective permeabilities are plotted vs- the shale proportion and show permeabilities are plotted vs- the shale proportion and show that a power averaging process would yield a good estimate much more accurate than either the arithmetic average (power 1) or geometric average (power 0) traditionally used. Connections with percolation theory results are indicated.}, 
  apa = {Journel, A. G., Deutsch, C., \& Desbarats, A. J. (1986, January). Power averaging for block effective permeability. In SPE California Regional Meeting. Society of Petroleum Engineers.
One of the most pervasive problems in the description of an heterogeneous medium is the problem of averaging from one scale to another. A medium property is observed at one scale on a particular property is observed at one scale on a particular support (volume) of measurement, but the value of that property ``averaged" over a different volume size at a different location is needed.

By ``average" it is meant the unique value of the latter volume that could replace the set of all smaller measurements that could be obtained and processed within it, were there no limitations of processed within it, were there no limitations of resolution, money, and/or CPU time.

For example, actual measurements can be performed on a small support such as permeabilities on performed on a small support such as permeabilities on core plugs and the effective permeability of a simulation block is required.

If the averaging process of the particular variable under study were known the problem would be much alleviated. For example, the average porosity of a volume is simply the arithmetic average of the porosities of all the samples that constitute it. porosities of all the samples that constitute it. The same arithmetic averaging process holds true for additive variables such as saturations and more generally grades or volume/weight percentages of various phases.

Unfortunately, many other medium characteristics are not additive, i.e. the corresponding averaging process is not a mere arithmetic average and, worse, process is not a mere arithmetic average and, worse, most often it is unknown. For example, the average or effective permeability of a block remains unknown even if all the thousands of core plugs that constitute it were accessible for analysis: it is neither the arithmetic average nor the geometric average of these core permeabilities, although in many situations practice has adopted the second averaging process. process. Compounding the problem is the fact that not all plugs (or supports) constituting the block are plugs (or supports) constituting the block are available for measurements. This second problem calls for interpolation or the unknown plug values from neighboring known values. Unfortunately, interpolating values before knowing how they average is like putting the cart before the horse and it does not matter how fancy the cart is, whether called kriging or splines. The key problem is the averaging process not the choice of the interpolation process.

Another word of caution should be addressed to the people in charge of data gathering, geologists and geophysicists. The goals of planning and monitoring production are in many ways fundamentally different from those of exploration.}
}



% ------------------------------------ %
















% ----------------- K ------------------ %

% King, MJ 1997
@inproceedings{king1997flow,
  title={Flow simulation of geologic models},
  author={King, Michael J and Mansfield, Mark and others},
  booktitle={SPE Annual Technical Conference and Exhibition},
  year={1997},
  organization={Society of Petroleum Engineers}, 
  keywords={}, 
  annote={},
  apa={King, M. J., \& Mansfield, M. (1997, January). Flow simulation of geologic models. In SPE Annual Technical Conference and Exhibition. Society of Petroleum Engineers.},
  ams={}
}


% Klausen
@article{klausen2012convergence,
  title={Convergence of multipoint flux approximations on General Grids and Media.},
  author={Klausen, Runhild A and Stephansen, Annette F},
  journal={International Journal of Numerical Analysis \& Modeling},
  volume={9},
  number={3},
  year={2012}, 
  apa={}, 
  annote={}, 
  keywords={MPFA, convergence},  
  ams={}
}



@article{klausen2006convergence,
  title={Convergence of multipoint flux approximations on quadrilateral grids},
  author={Klausen, Runhild A and Winther, Ragnar},
  journal={Numerical Methods for Partial Differential Equations: An International Journal},
  volume={22},
  number={6},
  pages={1438--1454},
  year={2006},
  publisher={Wiley Online Library}, 
  apa={}, 
  annote={}, 
  keywords={MPFA, convergence}, 
  ams={}
}


@article{klausen2006robust,
  title={Robust convergence of multi point flux approximation on rough grids},
  author={Klausen, Runhild A and Winther, Ragnar},
  journal={Numerische Mathematik},
  volume={104},
  number={3},
  pages={317--337},
  year={2006},
  publisher={Springer}, 
  apa={Klausen, R. A., \& Winther, R. (2006). Robust convergence of multi point flux approximation on rough grids. Numerische Mathematik, 104(3), 317-337.}, 
  annote={}, 
  keywords={MPFA, convergence}, 
  ams={}
}




% King, 1989 (Renormalization)
@article{king1989use,
	Author = {King, PR},
	Keywords = {Renormalization},
	Journal = {Transport in porous media},
	Number = {1},
	Pages = {37--58},
	Publisher = {Springer},
	Title = {The use of renormalization for calculating effective permeability},
	Volume = {4},
	Year = {1989}, 
	Annote = {If the permeability fluctuations are small then perturbation theory or effective medium theory (EMT) give reliable estimates of the effective permeability. However, for systems with a more severe permeability variation or for those with a finite fraction of nonreservoir rock all the simple estimates are invalid as well as EMT and perturbation theory.  Conventional simulation here refers to finite difference (or element) techniques for solving the single phase pressure equation. This requires the pressure and permeability at every grid point to be stored. Hence, these methods are limited in their resolution by the amount of data that can be stored in core.  The renormalization method involves averaging over small regions of the reservoir first to form a new ``averaged permeability'' distribution with a lower variance than the original. This pre-averaging may be repeated until a stable estimate is found.}}



% Kippe 2008 
@article{kippe2008comparison,
	Author = {Kippe, Vegard and Aarnes, J{\o}rg E and Lie, Knut-Andreas},
	Keywords = {elliptic pde},
	Annote = { }, 
	Journal = {Computational Geosciences},
	Number = {3},
	Pages = {377--398},
	Publisher = {Springer},
	Title = {A comparison of multiscale methods for elliptic problems in porous media flow},
	Volume = {12},
	Year = {2008}}




% Kitanidis
@article{kitanidis1990effective,
  title={Effective hydraulic conductivity for gradually varying flow},
  author={Kitanidis, Peter K},
  journal={Water Resources Research},
  volume={26},
  number={6},
  pages={1197--1208},
  year={1990},
  publisher={Wiley Online Library}, 
  Annote = { },
  Keywords = {Renormalization}
}




% Kumar
@inproceedings{kumar2004reservoir,
  title={Reservoir simulation of CO2 storage in deep saline aquifers},
  author={Kumar, Ajitabh and Noh, Myeong and Pope, GA and Sepehrnoori, K and Bryant, Steven and Lake, LW},
  booktitle={SPE/DOE Symposium on Improved Oil Recovery},
  year={2004},
  organization={OnePetro}, 
  annote={Modeling anisotropic flow through heterogenous porous media for CO2 sequestration.}
}


% ------------------------------------ %
















% ----------------- L ------------------ %

% Lambers
@article{lambers2008accurate,
	Annote = {VCMP DEVELOPED.  Constructs   a  2D local  finite volume scheme for  single phase flow using variable  multi-point transmissibility calculations. The scheme maximizes compactness of the stencil by adaptively choosing weights of six neighbors of a target face.  Numerical tests how the method significantly improves accuracy compared to other local methods and local--global method.  Results show },
	Author = {Lambers, James V. and Gerritsen, Margot G. and Mallison, Bradley T.},
	Keywords = {VCMP, 2D Local-global, CCAR, grid adaptivity, 2d VCMP},
	Journal = {Computational Geosciences},
	Notes = {VCMP applies to adaptive grids which are effective way to reduce upscaling errors and improved representation of connected flow paths in highly variable formations. Constructs VCMP, a local MPFA method that accommodates full-tensor anisotropy, but remains as close to a TPFA as possible. The method allows stencil to vary in both size and shape, reverting to a TPFA when accuracy is sufficiently high.},
	Number = {3},
	Pages = {399-416},
	Publicationstatus = {Published},
	Publisher = {Springer},
	Title = {Accurate local upscaling with variable compact multipoint transmissibility calculations},
	Volume = {12},
	Year = {2008}}



% Lambers
@inproceedings{lambers2009multiphase,
	Author = {Lambers, James Vincent and Gerritsen, Margot Gerritsen and Fragola, Daniele and others},
	Keywords = {},
	Annote = { },
	Booktitle = {SPE Reservoir Simulation Symposium},
	Organization = {Society of Petroleum Engineers},
	Title = {Multiphase, 3{D} flow simulation with integrated upscaling, {MPFA} discretization, and adaptivity},
	Year = {2009}}




% Lee
@article{lee2002implementation,
	Author = {Lee, Seong H and Tchelepi, Hamdi A and Jenny, Patrick and DeChant, Larry J and others},
	Keywords = {},
	Annote = { },
	Journal = {SPE Journal},
	Number = {03},
	Pages = {267--277},
	Publisher = {Society of Petroleum Engineers},
	Title = {Implementation of a flux-continuous finite-difference method for stratigraphic, hexahedron grids},
	Volume = {7},
	Year = {2002}}




% Leveque
@book{leveque2002finite,
	Author = {LeVeque, Randall J},
	Keywords = {FVM, hyperbolic},
	Annote = { },
	Publisher = {Cambridge university press},
	Title = {Finite volume methods for hyperbolic problems},
	Volume = {31},
	Year = {2002}}




% Lie, K. (2019). An Introduction to Reservoir Simulation Using MATLAB/GNU Octave: User Guide for the MATLAB Reservoir Simulation Toolbox (MRST). Cambridge: Cambridge University Press. doi:10.1017/9781108591416
@book{lie2019mrst, 
author={Lie, Knut-Andreas}, 
title={An Introduction to Reservoir Simulation Using MATLAB/GNU Octave: User Guide for the MATLAB Reservoir Simulation Toolbox (MRST)}, 
DOI={10.1017/9781108591416}, 
publisher={Cambridge University Press}, 
year={2019}, 
Keywords = {MRST},
Annote = { }
}


% Lie
@article{lie2012open,
  title={Open-source MATLAB implementation of consistent discretisations on complex grids},
  author={Lie, Knut--Andreas and Krogstad, Stein and Ligaarden, Ingeborg Skjelkv{\aa}le and Natvig, Jostein Roald and Nilsen, Halvor M{\o}ll and Skaflestad, B{\aa}rd},
  journal={Computational Geosciences},
  volume={16},
  number={2},
  pages={297--322},
  year={2012},
  publisher={Springer}
}

% Li 
@article{li2010anisotropic,
  title={An anisotropic mesh adaptation method for the finite element solution of heterogeneous anisotropic diffusion problems},
  author={Li, Xianping and Huang, Weizhang},
  journal={Journal of Computational Physics},
  volume={229},
  number={21},
  pages={8072--8094},
  year={2010},
  publisher={Elsevier}, 
  Annote={Li, X., \& Huang, W. (2010). An anisotropic mesh adaptation method for the finite element solution of heterogeneous anisotropic diffusion problems. Journal of Computational Physics, 229(21), 8072-8094.}, 
  keywords={FEM, anisotropic adaption}
}


@inproceedings{liu2016comparison,
  title={Comparison of averaging methods for interface conductivities in one-dimensional unsaturated flow in layered soils},
  author={Liu, Ruowen and Welfert, Bruno and Houston, Sandra},
  booktitle={International Symposium on Stratified Flows},
  volume={1},
  number={1},
  year={2016}, 
  annote={Which average is best for interface permeabilities?}
}


% ------------------------------------ %
















% ----------------- M ------------------ %

% Mallison
@inproceedings{mallison2006nonlinear,
  title={Nonlinear two-point flux approximations for simulating subsurface flows with full-tensor anisotropy},
  author={Mallison, BT and Chen, Y and Durlofsky, LJ},
  booktitle={ECMOR X-10th European Conference on the Mathematics of Oil Recovery},
  year={2006}, 
  Keywords = {tpfa, anisotropy},
	Annote = { }
}



% McKenna - Waste disposal
@article{mckenna2003modeling,
  title={Modeling dispersion in three-dimensional heterogeneous fractured media at Yucca Mountain},
  author={McKenna, Sean A and Walker, Douglas D and Arnold, Bill},
  journal={Journal of Contaminant Hydrology},
  volume={62},
  pages={577--594},
  year={2003},
  publisher={Elsevier},
  keywords = {Waste disposal;contaminant diffusion}
}

% Mittal
@article{mittal2005immersed,
	Author = {Mittal, Rajat and Iaccarino, Gianluca},
	Keywords = {},
	Annote = { },
	Journal = {Annu. Rev. Fluid Mech.},
	Pages = {239--261},
	Publisher = {Annual Reviews},
	Title = {Immersed boundary methods},
	Volume = {37},
	Year = {2005}}




% Mlacnik
@article{mlacnik2006unstructured,
  title={Unstructured grid optimization for improved monotonicity of discrete solutions of elliptic equations with highly anisotropic coefficients},
  author={Mlacnik, Martin J and Durlofsky, Louis J},
  journal={Journal of Computational Physics},
  volume={216},
  number={1},
  pages={337--361},
  year={2006},
  publisher={Elsevier},
  apa={Mlacnik, M. J., \& Durlofsky, L. J. (2006). Unstructured grid optimization for improved monotonicity of discrete solutions of elliptic equations with highly anisotropic coefficients. Journal of Computational Physics, 216(1), 337-361.}, 
  keywords={}, 
  annote={}, 
  abstract={Multipoint flux approximation (MPFA) techniques are commonly applied for discretizing the porous media flow equations within the context of finite volume numerical procedures. Although these methods can be applied to heterogeneous, anisotropic systems on generally unstructured grids, the inverse of the resulting linear operator can suffer from a loss of monotonicity at high permeability anisotropy ratios, resulting in spurious oscillations of the pressure solution. The purpose of this paper is to develop a method for optimizing unstructured grids in two and three dimensions such that the monotonicity behavior of the MPFA technique is significantly improved. The method employs anisotropic triangulation and can be readily combined with permeability upscaling procedures. Results are presented for a variety of examples and the technique is shown to perform well on problems involving complex grid point distributions and heterogeneous permeability fields with strong anisotropy ratios (of O(1 0 0)). Oscillation-free pressure solutions are achieved in all cases considered, even for examples in which the original grid shows large oscillations.}
}




% Mohammadnia, 2017 (monotonicity)
@article{mohammadnia2017monotonicity,
	Annote = {In order to match pressure transient data, a simulator must be able to calculate well bottomhole pressures highly accurately. Our objective is to develop discretizations that can be used to develop a thermal compositional simulator sufficiently accurate for well testing purposes. By sufficient accuracy, we mean that the wellbore pressure as a function of time and its derivative computed from the pressure by a standard finite difference approximation would match well the pressure and its derivative obtained from an analytical solution if they were available. To do so, we develop a new near-well local grid refinement within a base Cartesian grid and design discretization schemes using multipoint flux approximations (MPFA's) to obtain a numerical solution sufficiently accurate for well-test analysis. \textbf{We establish conditions that guarantee monotonicity for the specific MPFA schemes used. Although monotonicity in and of itself does not guarantee that pressure solutions will not exhibit nonphysical oscillations, monotone schemes do avoid decoupling of the solution which is a major cause of nonphysical oscillations in the pressure solution. Because we use local grid refinement within a Cartesian grid, there exist no previous results on monotonicity that apply directly to our discrete system. Thus, in this work, we derive criteria for discrete monotonicity for the proposed grid system, and then investigate the monotonicity regions of different MPFA methods as functions of permeability anisotropy and grid geometry properties}. To investigate the effects of the violation of a discrete maximum principle, we conduct tests on homogeneous and heterogeneous media, for both isotropic and anisotropic permeability fields.},
	Author = {Mohammadnia, Reza and Reynolds, Albert C and Forouzanfar, Fahim},
	Keywords = {Monotonicity, MPFA, oscillations in the pressure solution, decoupling of the solution},
	Journal = {Journal of Petroleum Science and Engineering},
	Pages = {707--728},
	Publisher = {Elsevier},
	Title = {Monotonicity conditions for {MPFA} methods for a numerical well testing reservoir simulator},
	Volume = {158},
	Year = {2017}}


% Monteagudo
@article{monteagudo2004control,
  title={Control-volume method for numerical simulation of two-phase immiscible flow in two-and three-dimensional discrete-fractured media},
  author={Monteagudo, JEP and Firoozabadi, Abbas},
  journal={Water resources research},
  volume={40},
  number={7},
  year={2004},
  publisher={Wiley Online Library}, 
  keywords={},
  annote={}, 
  ams={},
  apa={Monteagudo, J. E. P.,\& Firoozabadi, A. (2004). Control?volume method for numerical simulation of two?phase immiscible flow in two?and three?dimensional discrete?fractured media. Water resources research, 40(7).}  
}








% M{\o}yner
@article{moyner2014multiscale,
	Annote = {This paper will focus on multi-scale methods for computing pressure and fluxes on models that are realistic with the respect to geometrical description and petrophysical heterogeneity.  A large number of multi-scale finite-volume methods have been developed over the past decade to compute conservative approximations to multiphase flow problems in heterogeneous porous media. In particular, several iterative and algebraic multi-scale frameworks that seek to reduce the fine-scale residual towards machine precision have been presented. Common for all such methods is that they rely on a dual-primal coarse partition, which makes it challenging to extend them to stratigraphic and unstructured grids. Herein, we propose a general idea for how one can formulate multi-scale finite-volume methods using only a primal coarse partition. To this end, we use two key ingredients that are computed numerically: (i) elementary functions that correspond to flow solutions used in transmissibility upscaling, and (ii) partition- of-unity functions used to combine elementary functions into basis functions. We exemplify the idea by deriving a multi-scale two-point flux-approximation (MsTPFA) method, which is robust with regards to strong heterogeneities in the permeability field and can easily handle general grids with unstructured fine- and coarse-scale connections. The method can easily be adapted to arbitrary levels of coarsening, and can be used both as a standalone solver and as a preconditioner. Several numerical experiments are presented to demonstrate that the MsTPFA method can be used to solve elliptic pressure problems on a wide variety of geological models in a robust and efficient manner.},
	Author = {M{\o}yner, Olav and Lie, Knut-Andreas},
	Keywords = {MS-TPFA (Multi-scale TPFA)},
	Journal = {Journal of Computational Physics},
	Pages = {273--293},
	Publisher = {Elsevier},
	Title = {A multiscale two-point flux-approximation method},
	Volume = {275},
	Year = {2014}}





% Mostaghimi
@article{mostaghimi2015anisotropic,
  title={Anisotropic mesh adaptivity and control volume finite element methods for numerical simulation of multiphase flow in porous media},
  author={Mostaghimi, Peyman and Percival, James R and Pavlidis, Dimitrios and Ferrier, Richard J and Gomes, Jefferson LMA and Gorman, Gerard J and Jackson, Matthew D and Neethling, Stephen J and Pain, Christopher C},
  journal={Mathematical Geosciences},
  volume={47},
  number={4},
  pages={417--440},
  year={2015},
  publisher={Springer}
}


% ------------------------------------ %
















% ----------------- N ------------------ %

% Negara
@article{negara2015multiphase,
  title={Multiphase flow simulation with gravity effect in anisotropic porous media using multipoint flux approximation},
  author={Negara, Ardiansyah and Salama, Amgad and Sun, Shuyu},
  journal={Computers \& Fluids},
  volume={114},
  pages={66--74},
  year={2015},
  publisher={Elsevier}, 
  annote={},
  keywords={Multiphase, gravity effect, MPFA, anisotropic, porous media}
}





% Nilsson 2005 (Gridding, CCAR)
@inproceedings{nilsson2005novel,
	Author = {Nilsson, J. and Gerritsen, M. and Younis, R. and others},
	Keywords = {CCAR, grid adaptivity, anisotropic},
	Annote = {Development of CCAR for anisotropy. },
	Booktitle = {SPE reservoir simulation symposium},
	Organization = {Society of Petroleum Engineers},
	Title = {A novel adaptive anisotropic grid framework for efficient reservoir simulation},
	Year = {2005}}


% Nordbotten, 
@article{nordbotten2005discretization,
  title={Discretization on quadrilateral grids with improved monotonicity properties},
  author={Nordbotten, Jan Martin and Eigestad, Geir Terje},
  journal={Journal of computational physics},
  volume={203},
  number={2},
  pages={744--760},
  year={2005},
  publisher={Elsevier}, 
  Annote={},
  Keywords={}
}



% Nordbotten, 
@article{nordbotten2005monotonicity,
  title={Monotonicity conditions for control volume methods on uniform parallelogram grids in homogeneous media},
  author={Nordbotten, JM and Aavatsmark, Ivar},
  journal={Computational Geosciences},
  volume={9},
  number={1},
  pages={61--72},
  year={2005},
  publisher={Springer}, 
   Annote={},
   apa={},
  Keywords={monotonicity, control volume}
}



% Nordbotten, Aavatsmark 2007  Monotoncity and MPFA
@article{nordbotten2007monotonicity,
	Annote = {MPFA methods found to suffer from non-monotonicity, that is pressure solution may contain non-physical oscillations.  For homogeneous problems described by a full permeability tensor, it is shown that FV scheme with a similarly varying stencil is optimal in terms of monotonicity amongst a general class of schemes that are linearly exact.},
	Author = {Nordbotten, Jan M and Aavatsmark, Ivar and Eigestad, GT},
	Keywords = {MPFA, monotonicity, FVM, flexible stencil, control volume},
	Journal = {Numerische Mathematik},
	Number = {2},
	Pages = {255--288},
	Publisher = {Springer},
	Title = {Monotonicity of control volume methods},
	Volume = {106},
	Year = {2007}}

% ------------------------------------ %
















% ----------------- P ------------------ %

% Peaceman
@article{peaceman1997effective,
  title={Effective transmissibilities of a gridblock by upscaling-comparison of direct methods with renormalization},
  author={Peaceman, Donald W and others},
  journal={SPE Journal},
  volume={2},
  number={03},
  pages={338--349},
  year={1997},
  publisher={Society of Petroleum Engineers}, 
  keywords={transmissibility upscaling}, 
  annote={Previous methods for upscaling fine-scale permeability data for use in reservoir simulation produce effective permeabilities for each simulator gridblock. Since transmissibilities between neighboring gridblocks are required by the simulator, it is better to determine six "half-block transmissibilities" for each gridblock. These can be calculated directly by solving the finite-difference equations for pressure in each of the six half-blocks, wherein uniform pressures are applied at two opposite faces, and no-flow conditions are applied at the other four faces.  Alternatively, in order to reduce computation time, renormalization has been proposed for calculating the effective permeabilities of the gridblocks. These previous proposals ignore anisotropy and require that the elemental blocks and each coarse block be rectangular. In this paper, renormalization is modified to avoid these disadvantages. Renormalization is most easily implemented if there are NNN elemental blocks within each coarse block, where N is a power of 2. A more flexible variation of renormalization is also presented which reduces this restriction somewhat. However, the direct method is much more flexible with regard to the number of elemental blocks in each direction.  While renormalization might be expected to be faster than the direct method, a comparison of running times shows that a highly efficient iterative solver for the direct method is somewhat faster than renormalization.}  
}


% Pickup
@article{Pickup1994,
	Annote = {Main equation holds at upscale.  The same equation can be used to solve the coarse-scale pressure equation with effective permeability replacing fine-scale tensor and $p$ replaced by $p^c$.  Mathematical Geology},
	Author = {Pickup, G. E. and Ringrose, P. S. and Jensen, J. L. and Sorbie, K. S.},
	Keywords = {},
	Annote = { },
	Journal = {Mathematical Geology},
	Number = {2},
	Organization = {Heriot-Watt University},
	Pages = {227-250},
	Publisher = {Kluwer Academic Publishers-Plenum Publishers},
	Title = {Permeability tensors for sedimentary structures},
	Volume = {26},
	Year = {1994}}


% Pollock, 1988 
@article{pollock1988semianalytical,
	Author = {Pollock, David W},
	Keywords = {FDM},
	Annote = { },
	Journal = {Groundwater},
	Number = {6},
	Pages = {743--750},
	Publisher = {Wiley Online Library},
	Title = {Semianalytical computation of path lines for finite-difference models},
	Volume = {26},
	Year = {1988}}
	
	


% Pons, Adaptive Grid Refinement
@article{pons2016adaptive,
  title={Adaptive mesh refinement method. Part 1: Automatic thresholding based on a distribution function},
  author={Pons, K{\'e}vin and Ersoy, Mehmet},
  year={2016}, 
  Keywords = {AMR, Gridding}, 
  Annote = { }
}



@book{pyrcz2014geostatistical,
  title={Geostatistical reservoir modeling},
  author={Pyrcz, Michael J and Deutsch, Clayton V},
  year={2014},
  publisher={Oxford university press}, 
   Keywords = {AMR, Gridding}, 
  Annote = { }, 
  apa={Pyrcz, M. J., \& Deutsch, C. V. (2014). Geostatistical reservoir modeling. Oxford university press.}
}



% ------------------------------------ %






% ----------------- R ------------------ %

% Rating 2
@article{renard1997calculating,
	Annote = {The purpose of this article is to review the various methods used to calculate the equivalent permeability of a heterogeneous porous medium. It shows how equivalence is defined by using a criterion of flow or of the energy dissipated by viscous forces and explains the two different concepts of effective permeability and block permeability. The intention of this review is to enable the reader to use the various published techniques and to indicate in what circumstances they can be most suitably applied.},
	Author = {Renard, Ph and De Marsily, G},
	Keywords = {permeability upscaling effective permeability, block permeability},
	Journal = {Advances in water resources},
	Number = {5-6},
	Pages = {253--278},
	Publisher = {Elsevier},
	Title = {permeability upscaling: a review},
	Volume = {20},
	Year = {1997}}



% Romeu 1995
@article{romeu1995calculation,
	Author = {Romeu, RK and Noetinger, B},
	Keywords = {},
	Annote = { },
	Journal = {Water Resources Research},
	Number = {4},
	Pages = {943--959},
	Publisher = {Wiley Online Library},
	Title = {Calculation of internodal transmissivities in finite difference models of flow in heterogeneous porous media},
	Volume = {31},
	Year = {1995}}

% ------------------------------------ %
% trans- means "across"





% ----------------- S ------------------ %

% Saad - Matrix Methods
@book{saad2003iterative,
  title={Iterative methods for sparse linear systems},
  author={Saad, Yousef},
  volume={82},
  year={2003},
  publisher={SIAM}, 
  keywords={Matrix methods}
}


@article{sablok2008upscaling,
  title={Upscaling and discretization errors in reservoir simulation},
  author={Sablok, R and Aziz, Khalil},
  journal={Petroleum science and technology},
  volume={26},
  number={10-11},
  pages={1161--1186},
  year={2008},
  publisher={Taylor \& Francis}, 
annote={}, 
abstract={Reservoir simulation models are used as an everyday tool for reservoir management. The number of grid blocks in a simulation model is generally much smaller than the number of grid blocks in the geological model. The geological model is, therefore, regularly upscaled for building reasonably sized simulation models. Any upscaling causes a loss in detail and introduces errors. Understanding the nature of the errors that occur due to the upscaling step is important because it can provide a handle on the kind of upscaling to be performed and the optimum level of upscaling. Furthermore, understanding the interaction of gridding and upscaling errors is essential for building reliable simulation models. In this article, we analyze errors introduced due to the upscaling step.

First, the generality of the purely local single-phase upscaling is considered, and the errors introduced due to its use in complex multiphase flows are investigated. Three kinds of errors, namely total upscaling errors, discretization errors, and errors due to the loss of heterogeneity are defined, and their behavior as a function of the level of upscaling is studied for different types of permeability distributions. The reasons for apparently low total upscaling errors at high levels of upscaling are investigated. The efficacy of the geostatistical tools (variograms, QQ plots) for understanding the geological structure of the upscaled models, as compared to the reference model, is tested for different cases.

Second, the uncertainty introduced in the flow results due to the upscaling errors is studied in conjunction with the uncertainty incorporated through the introduction of geological variability in the ensemble of geological models. The behavior of upscaling errors and geological variability as a function of the level of upscaling is studied, and its possible impact on important reservoir management decisions is analyzed. It is shown that the uncertainty models of cumulative oil profiles, obtained from flow results of highly upscaled models can contain significant uncertainties. These uncertainties are a result of upscaling errors and, therefore, cannot be considered to represent only geological uncertainty, which is captured by the introduction of geological variability in the ensemble of the geological models.}, 
apa={Sablok, R., \& Aziz, K. (2008). Upscaling and discretization errors in reservoir simulation. Petroleum science and technology, 26(10-11), 1161-1186.}, 
keywords={ finite difference, geostatistical models, heterogeneity, porous media modeling}
}




% Saez
@article{saez1989effective,
  title={The effective homogeneous behavior of heterogeneous porous media},
  author={Saez, Avelino E and Otero, Carlos J and Rusinek, Isak},
  journal={Transport in porous media},
  volume={4},
  number={3},
  pages={213--238},
  year={1989},
  publisher={Springer}, 
  keywords={}, 
  annote={Theory of homogenization and using the same model for coarse scale as fine scale.}
}


% Samier
@inproceedings{samier1990finite,
  title={A finite element method for calculating transmissibilities in n-point difference equations using a non-diagonal permeability tensor},
  author={Samier, P},
  booktitle={ECMOR II-2nd European Conference on the Mathematics of Oil Recovery},
  year={1990}, 
  annote={Petroleum reservoirs are always heterogeneous. Averaging techniques consist mostly of defining an equivalent homogeneous permeability tensor for a given heterogeneous porous medium whose absolute permeability is a space-dependent function. The equivalent permeability tensor is generally symmetric but non-diagonal: three unusual off-diagonal terms, Kxy, Kxz, and Kyz, are to be considered in addition to the standard diagonal permeability terms in the x, y, and z directions. Non-diagonal tensors arise also in a mesh whose axes are not the principal directions of the permeability tensor: an application for horizontal well simulation is briefly presented.}, 
  abstract={Petroleum reservoirs are always heterogeneous. Averaging techniques consist mostly of defining an equivalent homogeneous permeability tensor for a given heterogeneous porous medium whose absolute permeability is a space-dependent function. The equivalent permeability tensor is generally symmetric but non-diagonal: three unusual off-diagonal terms, Kxy, Kxz, and Kyz, are to be considered in addition to the standard diagonal permeability terms in the x, y, and z directions. Non-diagonal tensors arise also in a mesh whose axes are not the principal directions of the permeability tensor: an application for horizontal well simulation is briefly presented.},
  keywords={}
}



% Sammon
@inproceedings{sammon2003dynamic,
	Author = {Sammon, Peter H and others},
	Keywords = {Grid refinement, Gridding, Adaptive Grid}, 
	Annote = { },
	Booktitle = {SPE reservoir simulation symposium},
	Organization = {Society of Petroleum Engineers},
	Title = {Dynamic grid refinement and amalgamation for compositional simulation},
	Year = {2003}}
	% amalgamation = combining or uniting




% Scheibe
@article{scheibe1998scaling,
  title={Scaling of flow and transport behavior in heterogeneous groundwater systems},
  author={Scheibe, Timothy and Yabusaki, Steven},
  journal={Advances in Water Resources},
  volume={22},
  number={3},
  pages={223--238},
  year={1998},
  publisher={Elsevier}, 
  Annote = { },
  Keywords = {scale averaging permeability}
}

% Shih - Constrained regularization
@article{shih2007constrained,
  title={Constrained regularization: Hybrid method for multivariate calibration},
  author={Shih, Wei-Chuan and Bechtel, Kate L and Feld, Michael S},
  journal={Analytical chemistry},
  volume={79},
  number={1},
  pages={234--239},
  year={2007},
  publisher={ACS Publications}
}




% Society of Petroleum Engineers (SPE)
@online{spe,
  author = {{Society of Petroleum Engineers}},
  title = {{Conceptual Visualization of the upscaling process. }},
  year = 2013,
  url = {https://petrowiki.org/images/2/25/Vol5_Page_1423_Image_0001.png},
  urldate = {2019-08-20}
}


% Srivastava
@article{srivastava1994overview,
  title={An overview of stochastic methods for reservoir characterization},
  author={Srivastava, R Mohan},
  year={1994},
  publisher={AAPG Special Volumes}
}



%  Stuben, 1983
@article{stuben1983algebraic,
	Author = {St{\"u}ben, Klaus},
	Keywords = {Algebraic multigrid},
	Annote = { }, 
	Journal = {Applied mathematics and computation},
	Number = {3},
	Pages = {419--451},
	Publisher = {Elsevier},
	Title = {Algebraic multigrid (AMG): experiences and comparisons},
	Volume = {13},
	Year = {1983}}

% ------------------------------------ %
















% ----------------- T ------------------ %

% Terekhov and Mallison, 2017
@article{terekhov2017cell,
	Author = {Terekhov, Kirill M and Mallison, Bradley T and Tchelepi, Hamdi A},
	Journal = {Journal of Computational Physics},
	Keywords = {Finite volume method, Discrete maximum principle, Positivity preserving property, Anisotropic diffusion equation, Interpolation method, Unstructured mesh},
	Pages = {245--267},
	Publisher = {Elsevier},
	Title = {Cell-centered nonlinear finite-volume methods for the heterogeneous anisotropic diffusion problem},
	Volume = {330},
	Year = {2017}, 
	Annote = {We present two new cell-centered nonlinear finite-volume methods for the heterogeneous, anisotropic diffusion problem. The schemes split the interfacial flux into harmonic and transversal components. Specifically, linear combinations of the transversal vector and the co-normal are used, which leads to significant improvements in terms of the mesh-locking effects. The harmonic component of the flux is represented using a conventional monotone two-point flux approximation; the component along the parameterized direction is treated nonlinearly to satisfy either positivity of the solution as in [29], or the discrete maximum principle as in [9]. In order to make the method purely cell-centered, we derive a homogenization function that allows for seamless interpolation in the presence of heterogeneity following a strategy similar to [46]. The performance of the new schemes is compared with existing multi-point flux approximation methods [3], [5]. The robustness of the scheme with respect to the mesh-locking problem is demonstrated using several challenging test cases.}}




% Trangenstein 2001
@article{trangenstein2001multi,
	Annote = {Many enhanced oil recovery processes in reservoir engineering involve localized phenomena that could be due to several features, such as injection fronts, wells, or reservoir heterogeneity. In order to reach sufficient accuracy in field-scale simulation, the localized phenomena need to be resolved and modeled in appropriate \hl{scale-dependent ways}. Our approach to treating the localized phenomena is to use high-resolution discretization methods in combination with dynamic adaptive mesh refinement(AMR). The purpose of adaptive mesh refinement is to concentrate the computational work near the regions of interest in the displacement processes, which may evolve constantly in space. Adaptive mesh refinement requires appropriate techniques for data communication in a hierarchy of dynamically adaptive mesh. The selection of appropriate scaling rules, as well as computationally efficient data structures, is essential to the success of the overall method.},
	Author = {Trangenstein, John A and Bi, Zhuoxin},
	Keywords = {Adaptive Grid Refinement},
	Publisher = {Citeseer},
	Title = {Multi-scale iterative techniques and adaptive mesh refinement for miscible displacement simulation},
	Year = {2001}}

% ------------------------------------ %







% ----------------- V ------------------ %

@article{varga1966discrete,
  title={On a discrete maximum principle},
  author={Varga, Richard S},
  journal={SIAM Journal on Numerical Analysis},
  volume={3},
  number={2},
  pages={355--359},
  year={1966},
  publisher={SIAM}, 
  Annote={Varga, R. S. (1966). On a discrete maximum principle. SIAM Journal on Numerical Analysis, 3(2), 355-359.}
}


% Crossref: (2010) An anisotropic mesh adaptation method for the finite element solution of heterogeneous anisotropic diffusion problems. Journal of Computational Physics 229:21, 8072-8094.












% ----------------- W ------------------ %




% Wallstrom (Durlofsky, Hou, Christie, Sharp
@inproceedings{wallstrom1999application,
  title={Application of a new two-phase upscaling technique to realistic reservoir cross sections},
  author={Wallstrom, TC and Hou, S and Christie, MA and Durlofsky, LJ and Sharp, DH and others},
  booktitle={SPE Reservoir Simulation Symposium},
  year={1999},
  organization={Society of Petroleum Engineers}, 
  Keywords = {},
	Annote = {constant pressure condition at the sub-domain boundary tends to overestimate flow contributions from high permeability areas.}
}





% Warren and Price, 1961
@article{warren1961flow,
  title={Flow in heterogeneous porous media},
  author={Warren, JE and Price, HS and others},
  journal={Society of Petroleum Engineers Journal},
  volume={1},
  number={03},
  pages={153--169},
  year={1961},
  publisher={Society of Petroleum Engineers}, 
  keywords={no-flow boundary conditions, sealed-sides}, 
  annote={Imposed a constant pressure gradient in a selected direction of flow by specifying a pressure of 1 on the inflow face and a pressure of 0 on the outflow face. By allowing no flow to pass through the sides of the cell, all fluxes are forced to go in the principal direction of flow. Therefore, this type of boundary conditions is often referred to as the no-flow or sealed-sides boundary conditions.}, 
  apa={}, 
  ams={}
}


% Watson, 2013
@book{watson2013contouring,
	Author = {Watson, Debbie},
	Keywords = {},
	Annote = { },
	Publisher = {Elsevier},
	Title = {Contouring: a guide to the analysis and display of spatial data},
	Volume = {10},
	Year = {2013}}


% Weiser
@article{weiser1988convergence,
  title={On convergence of block-centered finite differences for elliptic problems},
  author={Weiser, Alan and Wheeler, Mary Fanett},
  journal={SIAM Journal on Numerical Analysis},
  volume={25},
  number={2},
  pages={351--375},
  year={1988},
  publisher={SIAM}, 
  annote={}, 
  abstract={We consider linear, selfadjoint, elliptic problems with Neumann boundary conditions in rectangular domains. We demonstrate that with sufficiently smooth data, the discrete $L^2$-norm errors for tensor product block-centered finite differences in both the approximate solution and its first derivatives are second-order for all nonuniform grids. Extensions to non self adjoint and parabolic problems are discussed.}, 
  apa={Weiser, A., \& Wheeler, M. F. (1988). On convergence of block-centered finite differences for elliptic problems. SIAM Journal on Numerical Analysis, 25(2), 351-375.}, 
  keywords={convergence}, 
  ams={65}
}



% Wen 2006
@article{wen2006efficient,
	Annote = {Upscaling is often applied to coarsen detailed geological reservoir descriptions to sizes that flow simulators can accommodate. Adaptive local-global upscaling is a new and accurate methodology that incorporates global coarse-scale flow information into the boundary conditions used to compute upscaled quantities (e.g., coarse-scale transmissibilities). The procedure is iterated until a self-consistent solution is obtained. In this work, we extend this approach to 3D systems and introduce and evaluate procedures to decrease the computational demands of the method. This includes the use of purely local upscaling calculations for the initial estimation of coarse-scale transmissibilities and the use of reduced border regions during the iterations. This is shown to decrease the computational requirements of the reduced procedure significantly relative to the full methodology while impacting the accuracy very little. The performance of the adaptive local-global upscaling technique is evaluated for three different heterogeneous reservoir descriptions. The method is shown to provide a high degree of accuracy for relevant flow quantities. In addition, it is shown to be less computationally demanding and significantly more accurate than some existing extended local upscaling procedures.},
	Author = {Wen, Xian-Huan and Durlofsky, Louis J and Chen, Yuguang and others},
	Keywords = { },
	Journal = {Spe Journal},
	Number = {04},
	Pages = {443--453},
	Publisher = {Society of Petroleum Engineers},
	Title = {Efficient 3D implementation of local-global upscaling for reservoir simulation},
	Volume = {11},
	Year = {2006}}




% Wen - Grid generation
@article{wen2003upscaling,
  title={Upscaling of channel systems in two dimensions using flow-based grids},
  author={Wen, XH and Durlofsky, LJ and Edwards, MG},
  journal={Transport in Porous Media},
  volume={51},
  number={3},
  pages={343--366},
  year={2003},
  publisher={Springer}, 
  apa={Wen, X. H., Durlofsky, L. J., \& Edwards, M. G. (2003). Upscaling of channel systems in two dimensions using flow-based grids. Transport in Porous Media, 51(3), 343-366.}, 
  annote={}, 
  keywords = {amr, upscaling, channelized}, 
  abstract={A methodology for the gridding and upscaling geological systems characterized by channeling is presented. The overall approach entails the use of a flow-based gridding procedure for the generation of variably refined grids capable of resolving the channel geometry, a specialized full-tensor upscaling method to capture the effects of permeability connectivity, and the use of a flux-continuous finite volume method applicable to full tensor permeability fields and non-orthogonal grids. The gridding and upscaling procedures are described in detail and applied to several two-dimensional systems. Significant improvement in the accuracy of the coarse-scale models, relative to that obtained using uniform Cartesian coarse-scale models, is achieved in all cases. It is shown that, for some systems, improvement results from the use of the flow-based grid, while in other cases, the improvement is mainly due to the new upscaling method.}
}




%  Wen 2003
@article{wen2003use,
	Annote = {A procedure for the improved calculation of upscaled grid block permeability tensors on Cartesian grids is described and applied. The method entails using a border region of fine-scale cells surrounding the coarse block for which the upscaled permeability is to be computed.  The method provided enhanced accuracy relative to standard procedures because it introduces the effects of larger-scale permeability connectivity into calculating the upscaled block permeability tensor.    Overall findings indicate that border regions improve upon standard methods by better resolving the effects of permeability connectivity, BUT adding border regions does not lead to significantly more accurate coarse models when standard upscaling procedures adequately model the effects of permeability connectivity.  The degree of improvement varies for different models and flow quantities.  \hl{Using one bordering ring surrounding the target cell ($r = 1$) appears to suffice. Relatively little improvement was observed when a two-ring region was applied.}
  	},
	Author = {Wen, XH and Durlofsky, LJ and Edwards, MG},
	Journal = {Mathematical Geology},
	Keywords = {Border Regions, Extended Local procedure for permeability upscaling},
	Number = {5},
	Pages = {521--547},
	Publisher = {Springer},
	Title = {Use of border regions for improved permeability upscaling},
	Volume = {35},
	Year = {2003}}




% Wen 1996
@article{wen1996upscaling,
	Annote = {A review of the many existing techniques for conductivity upscaling. Coarse permeability is understood as not being a material property but dependent on the flow conditions within the block.    Upscaling is a process that transforms a grid of hydraulic conductivities defined at the scale of the measurements, into a coarser grid of block conductivity tensors amenable for input to a \hl{numerical flow simulator}. \hl{The need for upscaling stems from the disparity between the scales at which measurements are taken and the scale at which aquifers are discretized for the numerical solution of flow and transport}. The techniques for upscaling range from the simple averaging of the heterogeneous values within the block to sophisticated inversions, after the solution of the flow equation at the measurement scale within an area embedding the block being upscaled. All techniques have their own advantages and limitations. Recently, the definition of the geometry of the grid has been intimately linked to the upscaling problem; promising results have been obtained using elastic gridding. Also, recently, the need to perform Monte-Carlo analysis, involving many realizations of hydraulic conductivity, has steered the development of methods that generate directly the block conductivities in accordance with the rules of upscaling, yet conditional to the measurement data.},
	Author = {Wen, Xian-Huan and G{\'o}mez-Hern{\'a}ndez, J Jaime},
	Keywords = {Permeability Upscaling, elastic gridding},
	Journal = {Journal of Hydrology},
	Number = {1-2},
	Pages = {ix--xxxii},
	Publisher = {Elsevier},
	Title = {Upscaling hydraulic conductivities in heterogeneous media: An overview},
	Volume = {183},
	Year = {1996}}




% White
@inproceedings{white1987computing,
  title={Computing absolute transmissibility in the presence of fine-scale heterogeneity},
  author={White, CD and Horne, RN and others},
  booktitle={SPE symposium on reservoir simulation},
  year={1987},
  organization={Society of Petroleum Engineers}, 
  Keywords = {transmissibility upscaling},
	Annote = {This paper presents an algorithm to compute transmissibility when there is permeability heterogeneity and anisotropy at the subgrid scale. The need for a tensor representation of macroscopic transmis-sibility in order to scale properly the effects of microscale permeability variations is demonstrated. Methods for computing tensor transmissibility and the use of tensor transmissibility in reservoir simulation are described. The proposed method is applicable even when the variance of permeability is large and the principal directions of the transmissibility tensor are not aligned with the coordinate axes. Examples demonstrate that the general tensor scaling procedure can give accurate, efficient production estimates on a coarse grid.}
}









% Wu, 2009 effects
@article{wu2009effect,
  title={Effect of grid deviation on flow solutions},
  author={Wu, Xiao-Hui and Parashkevov, Rossen and others},
  journal={SPE journal},
  volume={14},
  number={01},
  pages={67--77},
  year={2009},
  publisher={Society of Petroleum Engineers}, 
  keywords={},
  annote={}
}




% Wu 2002
@article{wu2002analysis,
	Author = {Wu, Xiao-Hui and Efendiev, Y and Hou, Thomas Y},
	Keywords = {},
	Annote = {XLocal better results upscaling permeability},
	Journal = {Discrete and Continuous Dynamical Systems Series B},
	Number = {2},
	Pages = {185--204},
	Publisher = {AIMS PRESS},
	Title = {Analysis of upscaling absolute permeability},
	Volume = {2},
	Year = {2002}}

% ------------------------------------ %










% ----------------- X ------------------ %

@article{xu2017algebraic,
  title={Algebraic interface-based coarsening AMG preconditioner for multi-scale sparse matrices with applications to radiation hydrodynamics computation},
  author={Xu, Xiaowen and Mo, Zeyao},
  journal={Numerical Linear Algebra with Applications},
  volume={24},
  number={2},
  pages={e2078},
  year={2017},
  publisher={Wiley Online Library}
}









% ----------------- Y ------------------ %


% Yang, 2014
@book{yang2014introduction,
	Annote = {Textbook covering topics from computational math.  In particular, FDM, FVM, and FEM.},
	Author = {Yang, Xin-She},
	Keywords = {FVM, FEM, FDM},
	Publisher = {World Scientific Publishing Company},
	Title = {Introduction to computational mathematics},
	Year = {2014}}




% Yang, 2002
@article{yang2002boomeramg,
	Author = {Yang, Ulrike Meier and others},
	Keywords = {},
	Annote = { },
	Journal = {Applied Numerical Mathematics},
	Number = {1},
	Pages = {155--177},
	Publisher = {Elsevier},
	Title = {BoomerAMG: a parallel algebraic multigrid solver and preconditioner},
	Volume = {41},
	Year = {2002}}





% Younis, 2002
@inproceedings{younis2002method,
	Annote = {\hl{Inclusion of local and global static information can greatly improve computational cost while maintaining accurate flow response}.  A method for up-gridding geostatistical reservoir models is proposed. The method generates well-adapted coarse-grid grids for a given realization and target coarse-scale dimensions. While most current grid generation methods are based on local or global dynamic (flow rate dependent) measures, this paper shows that including local and global static information can greatly improve computational costs while maintaining accurate flow response. Static property information is analyzed to infer a measure of spatial variability regarding the relative influence of connected bodies on reservoir flow response. These bodies consist of well-connected permeability falling within a range of permeability values. Bodies may strongly impact local and global flow behavior; knowledge of the relative location of such bodies in a given realization may render an accurate coarse-scale representation of the fine-scale models in terms of flow response. An investigation of the method under a set of realistic synthetic 2D and 3D reservoirs is conducted. Sensitivity to the strength of geological continuity and flow boundary conditions are considered for various scale-up ratios. The results indicate accurate reproductions of the flow simulation response compared to fine-scale realizations. A comparative analysis of the method with well-established static methods is also conducted.},
	Author = {Younis, Rami M and Caers, Jef},
	Keywords = {gridding},
	Booktitle = {ECMOR VIII-8th European Conference on the Mathematics of Oil Recovery},
	Title = {A Method for Static-based Up-gridding},
	Year = {2002}}




% Zhang, 2017
@article{zhang2017two,
	Annote = {A novel Two-Step cell-centered Finite Volume Method (TSFVM) is developed in this work to discretize the heterogeneous and anisotropic pressure equation on triangular and quadrilateral grids in 2D and hexahedral and tetrahedral grids in 3D. Each grid cell's physical properties, such as permeability and porosity, are piece-wise constant. In the first step, the Galerkin Finite Element Method (FEM) is utilized to compute pressure solutions at all cell vertices. In the second step, pressure values at cell vertices derive continuous two-point flux stencils for cell faces. Mass conservation equations are then written for each cell to obtain a system of linear equations that can be solved for pressure at cell centers. Extensive numerical experiments are carried out to test the performance of our TSFVM. In particular, we compare TSFVM with the classical Multipoint Flux Approximation (MPFA-O) method as well as a more recently developed MPFA method with full pressure support called enhanced MPFA (eMPFA). The results show that the TSFVM compares well with eMPFA for challenging test cases for which MPFA-O breaks down. Specifically, and as a significant step forward, our TSFVM is robust for challenging problems involving heterogeneous and highly anisotropic permeability tensors when both MPFA-O and eMPFA suffer from unphysical oscillations. Finally, the numerical convergence study demonstrates that TSFVM has comparable convergence behavior to MPFA-O method for both homogeneous and discontinuous permeability fields.},
	Author = {Zhang, Wenjuan and Al Kobaisi, Mohammed},
	Keywords = {FVM, FEM, anisotropic pressure equation, general grids},
	Journal = {Advances in Water Resources},
	Pages = {231--248},
	Publisher = {Elsevier},
	Title = {A two-step finite volume method to discretize heterogeneous and anisotropic pressure equation on general grids},
	Volume = {108},
	Year = {2017}}
	
	
	
	
@article{zhang2017simplified,
  title={A simplified enhanced {MPFA} formulation for the elliptic equation on general grids},
  author={Zhang, Wenjuan and Al Kobaisi, Mohammed},
  journal={Computational Geosciences},
  volume={21},
  number={4},
  pages={621--643},
  year={2017},
  publisher={Springer}, 
  keywords={MPFA}, 
  annote={Multi-point flux approximation (MPFA) has proven to be a powerful tool for discretizing the diffusion equation on general grids with heterogeneous anisotropic permeability tensors and, hence, removing the O (1) error introduced by two-point flux approximation (TPFA) for non-K-orthogonal grids. However, it is well known that the classical MPFA-O suffers from monotonicity issues and strong unphysical oscillations can be present for highly anisotropic media.}
}





@article{zhang2008new,
  title={A new practical method for upscaling in highly heterogeneous reservoir models},
  author={Zhang, Pinggang and Pickup, Gillian E and Christie, Michael A and others},
  journal={SPE Journal},
  volume={13},
  number={01},
  pages={68--76},
  year={2008},
  publisher={Society of Petroleum Engineers}, 
  keywords={}, 
  annote={}
}




@article{zou2005regularization,
  title={Regularization and variable selection via the elastic net},
  author={Zou, Hui and Hastie, Trevor},
  journal={Journal of the royal statistical society: series B (statistical methodology)},
  volume={67},
  number={2},
  pages={301--320},
  year={2005},
  publisher={Wiley Online Library}, 
  keywords={}, 
  annote={}
}





% additional:

% https://www.ams.org/journals/mcom/0000-000-00/S0025-5718-2021-03671-6/

% Jean-Marie Mirebeau, Minimal stencils for discretizations of anisotropic PDEs preserving causality or the maximum principle, SIAM J. Numer. Anal. 54 (2016), no. 3, 1582–1611. MR 3504992, DOI 10.1137/16M1064854


% transmissivity = https://www.merriam-webster.com/dictionary/transmission
  annote={}, 
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  apa={}, 
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  ams={}