Matmídia Laboratory Department of Mathematics PUC-Rio

matmidia.mat.puc.rio.br

Matmídia Laboratory Department of Mathematics PUC-Rio

Sibgrapi 2009 Workshop on Petroleum Applications

PUC-Rio

Matmídia Laboratory

Department of Mathematics

PUC-Rio

Transfering Computer Graphics Academic Research

to the Oil Industry


Matmídia Laboratory

Matmídia Laboratory

Transfering Computer Graphics

Academic Research

to the Oil Industry


http://www.matmidia.org

Interface with the Oil Industry

Software Design

Matmídia Project

Cycle Development

Software Development

Real Data Validation

Company’s Dissemination

Sibgrapi 2009 Workshop on Petroleum Applications


http://www.matmidia.org

Presentation Topics

• Topological Data Structures

• Petbool

• Implicit Modeling

• GeosisBR

• Machine Learning

• KernelsisBR

• Meshless Fluid Dynamics

• Multifluid

Sibgrapi 2009 Workshop on Petroleum Applications


Presentation

http://www.matmidia.org

Among several projects we have developed

for Petrobras, we have chosen, for this

presentation, those placed at the

crossroads of our research today and our

software development for the oil industry

Sibgrapi 2009 Workshop on Petroleum Applications


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Presentation Topics

• Topological Data Structures

• Petbool

• Implicit Modeling

• Geosis

• Machine Learning

• Kernelsis

• Meshless Fluid Dynamics

• Multifluid

Sibgrapi 2009 Workshop on Petroleum Applications


Introduction

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The Matmídia Laboratory has been working

with Petrobras since 1991.

This partnership has been a source of

renewed challenges both to our academic

research as well as industrial projects.

The transfer of applied mathematics

generated technologies is done respecting

both academic requirements and industrial

secrecy constraints.

Sibgrapi 2009 Workshop on Petroleum Applications


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Topological Data Structures

• Morse Theory - 1930’s.

• Morse theory based topological data

structures -1992-2009.

• Splines based topological structures.

• Visualization based on topological data

structures.

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Topological Data Structures

A topological data structure builds an abstract structure

relating the connectivity of its core elements :

• vertices,

• edges,

• faces,

• 3D cells.

The most common instantiation of a topological data

structure is a geometrical object, volumes and

surfaces, for example.

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Topological Data Structures

One of the advantages of this kind of

structure is the possibility of building an

unified environment for the handling of

geometric objects and interactive

visualization.

Sibgrapi 2009 Workshop on Petroleum Applications


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Petbool: Stochastic Modeling of Geological Objects

The software Petbool is based on geological

objects and boolean stochastic simulation.

Its main feature is to build three dimensional

scenarios of equiprobable simulations of oil

reservoirs with data conditioned to the wells

and to the geological formation.

Sibgrapi 2009 Workshop on Petroleum Applications


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Petbool: Stochastic Modeling of Geological Objects

Based on geological forms specified by

their geometrical parameters.

The genetics units are the basic

Petbool's building blocks: channels,

lenses, domes, dunes, sigmoids and

sheets.

Sibgrapi 2009 Workshop on Petroleum Applications


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Petbool: Stochastic Modeling of Geological Objects

Fluvial Channel Simulation

297 wells

Potiguar Basin, Petrobras, Brazil

Sibgrapi 2009 Workshop on Petroleum Applications


Implicit Modeling

http://www.matmidia.org

This area of research in of geometric modeling

is based on boolean operations of implicit functions

f(x,y,z) = 0

Sibgrapi 2009 Workshop on Petroleum Applications


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GeoSis: Seismic Geobodies Reconstruction

A given interval is specified for a seismic

property.

The corresponding geobody boundary

isosurface is visualized and can be explored

in several ways.

The geobody topology is guaranteed.

Sibgrapi 2009 Workshop on Petroleum Applications


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GeoSis: Seismic Geobodies Reconstruction

3D reconstruction

from the seismic

amplitude.

Sibgrapi 2009 Workshop on Petroleum Applications


Machine Learning

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• Data

– A training set

{x i

, y i

} where x i

are parameters

and y i

are labels.

• Output function

– A classifying function

Fx

()= α i

l

∑ Kx ( i

,x)+ b

i =1

Sibgrapi 2009 Workshop on Petroleum Applications


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KernelSis: Machine Learning and Modeling Sismolithofacies

• Context

– To correlate volumetric seismic

multiattributes with petrophysical well

properties.

• Mathematical and Computational

Environment

– Kernel methods and support vector

machines in statistical learning

Sibgrapi 2009 Workshop on Petroleum Applications


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KernelSis: Machine Learning and Modeling Sismolithofacies

KernelSis

Facies

Identification

Sibgrapi 2009 Workshop on Petroleum Applications


Meshless Fluid Dynamics

http://www.matmidia.org

In its lagrangean formulation Navier-Stokes equations

for incompressible gravity flows are given by:

Sibgrapi 2009 Workshop on Petroleum Applications


Meshless Fluid Dynamics

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The convolution with a gaussian of each

component of the velocity and its

gradient are discretized.

Then using a state equation for the

pressure and this discretization, the

Navier-Stokes equations are solved by

the projection method.

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MultiFluid: Modeling Turbidite Lobes Process Formation

The next picture is a 2D version of the gravity

flow for a two-fase flow.

One of the theories is that turbidite lobes are formed

by avalanches followed by deposition of organic material

Sibgrapi 2009 Workshop on Petroleum Applications


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References

Topological Data Structures

Tavares, G. ; LOPES, H. . Handlebody Representation for Surfaces and Morse Operators. In: Curves and

Surfaces in Computer Vision Graphics III, 1992, Boston. Proceedings SPE of Curves and Surfaces in

Computer Vision Graphics III. Boston, Estados Unidos : SPIE, 1992.

Hélio Lopes. Uma Estrutura de Dados para Representação de Superfícies'. 1992. 0 f. Dissertação (Mestrado em

Matemática) - Pontifícia Universidade Católica do Rio de Janeiro.

Tavares, G. ; LOPES, H. ; CASTELO, A. . Topological Methods in Geometric Modeling. Rio de Janeiro: SBMAC,

1992.

Sinésio Pesco. Modelagem Spline em um Ambiente Topológico. 1993. 0 f. Dissertação (Mestrado em

Matemática) - Pontifícia Universidade Católica do Rio de Janeiro.

PESCO, S. ; Tavares, G. . Splines in a Topological Setting. In: Pierre-Jean Laurent; Alain Le Méhauté; Larry L.

Schumaker. (Org.). Curves and Surfaces in Geometric Design. Wellesley, MA, Estados Unidos: A. K.

Peters, Ltd., 1994, v. 1, p. 361-368.

Hélio Lopes. Construção e Desconstrução de Variedades de Dimensões 2 e 3. 1997. 0 f. Tese (Doutorado em

Matemática) - Pontifícia Universidade Católica do Rio de Janeiro.

Tavares, G. ; LOPES, Hélio . Structural Operators for Modeling 3-Manifolds. In: 4th Simposium on Solid

Modeling and Aplications, 1997, Atlanta. Fourth Symposium on Solid Modeling and Applications. New York

: ACM Press, 1997. p. 10-18.

Sinésio Pesco. Modelagem Geométrica com Espaços Singulares'. Departamento de Matemática. 1997. 0 f.

Tese (Doutorado em Matemática) - Pontifícia Universidade Católica do Rio de Janeiro.

Pesco, S ; Tavares, G. ; Lopes, H. . A stratification approach for modeling two-dimensional cell complexes.

Computers & Graphics, Inglaterra, v. 28, n. 2, p. 235-247, 2004.

Sibgrapi 2009 Workshop on Petroleum Applications


References

http://www.matmidia.org

Topological Data Structures

Luis Gustavo Nonato. Reconstrução Volumétrica de Objetos Dados por Seções Bidimensionais. 1998. 0 f. Tese

(Doutorado em Matemática) - Pontifícia Universidade Católica do Rio de Janeiro

Lopes, H. ; Nonato, L. G. ; Pesco, S.; Tavares, G. . Dealing with topological singularities in volumetric

reconstruction. . In: Curves and Surfaces Design, 2000, Saint-Malo. Curves and Surfaces Design.

Nashville, USA : Vanderbilt University Press, 2000. p. 229-238.

Nonato, L. ; Minghim, R. ; Oliveira, M. C. F. ; Tavares, G. . A Novel Approach for Delaunay 3D Reconstruction

with a Comparative Analysis in the Light of Applications. Computer Graphics Forum, v. 20, n. 2, p. 161-174,

2001.

MAIA, M. ; Tavares, G. ; LOPES, H. ; XAVIER, A. ; PESCO, S. . EXTRACTION OF TERRAIN DATA FROM

DIGITISED CONTOURS. International Journal of Shape Modeling, Singapura, v. 7, n. 1, p. 59, 2001.

LOPES, H. ; Tavares, G. ; XAVIER, A. ; MAIA, M. G. ; Pesco, S . HANDLEBODY REPRESENTATION FOR

SURFACES AND ITS APPLICATIONS TO TERRAIN MODELING. International Journal of Shape

Modeling, v. 9, n. 1, p. 61, 2003

Lopes, H ; Tavares, G. ; ROSSIGNAC, Jarek ; SAFANOVA, Alla ; SZYMPACK, A. . Edgebreaker: a simple

implementation for surfaces with handles. Computers & Graphics, Inglaterra, v. 27, n. 4, p. 553-567, 2003.

Vieira, Antonio W. ; LEWINER, Thomas ; VELHO, Luiz ; Lopes, Helio ; Tavares, Geovan . Stellar Mesh

Simplification Using Probabilistic Optimization. Computer Graphics Forum, v. 23, p. 825-838, 2004.

Lewiner, T. ; LOPES, H. ; Tavares, G. . Applications of Forman’s discrete Morse theory to topology visualization

and mesh compression. IEEE Transactions on Visualization and Computer Graphics, v. 10, n. 5, p. 499-

508, 2004

Tavares, G. ; MINGHIM, R. ; LEVKOWITZ, H. ; NONATO, L. G. ; WATANABE, L. ; SALVADOR, V. ; LOPES, H.

; PESCO, S. . Spider Cursor: a simple versatile interaction tool for data visualization and exploration. In: 3rd

international conference on Computer graphics and interactive techniques in Australasia and South East

Asia, 2005, Dunedin. GRAPHITE '05, 2005. v. 1. p. 307-313.

Sibgrapi 2009 Workshop on Petroleum Applications


References

http://www.matmidia.org

Topological Data Structures

Tavares, G. ; Lewiner, Thomas ; LOPES, H. . Towards Optimality in Discrete Morse Theory. Experimental

Mathematics, Estados Unidos, v. 12, n. 3, p. 271-285, 2003.

Thomas Lewiner. Complexos de Morse Discretos e Geométricos. 2005. Tese (Doutorado em Matemática) -

Pontifícia Universidade Católica do Rio de Janeiro.

Castro, Rener ; LEWINER, Thomas ; LOPES, Hélio ; Tavares, Geovan ; Bordignon, Alex . Statistical

optimization of octree searches. Computer Graphics Forum, v. 27, p. 1557-1566, 2008.

Rener Pereira de Castro. Otimização Estatística de Buscas Para Estruturas Hierárquicas Espaciais. 2008. Tese

(Doutorado em Matemática) - Pontifícia Universidade Católica do Rio de Janeiro.

Velho, Luiz ; Lopes, H. ; E. Medeiros ; Lewiner, Thomas ; Tavares, G . Topological Mesh Operators. Computer

Aided Geometric Design, 2009.

Sibgrapi 2009 Workshop on Petroleum Applications


References

http://www.matmidia.org

Petbool

Tavares, G. ; LANZARINI, W. ; FARIA, R. T. ; BARRETO, A. B. ; POLLETO, A. ; LOPES, Hélio ; PESCO, S. .

Modelagem Estocástica/ Booleana da Geometria Tridimensional de Reservatórios de Petróleo. In:

CONEXPO ARPEL 96, 1996, Rio de janeiro. CONEXPO ARPEL 96, 1996.

Tavares, G. ; LANZARINI, W. ; PESCO, S. ; LOPES, Hélio . Parametrização e Modelagem Geométrica de

Canais Fluviais da Formaçào Açu. In: I Seminário de Reservas e Reservatórios da Petrobras, 1996, Rio de

Janeiro. I Seminário de Reservas e Reservatórios da Petrobras, 1996.

Tavares, G. ; LANZARINI, W. ; FARIA, R. T. ; BARRETO, A. B. ; LOPES, Hélio ; PESCO, S. . Petbool:

Simulador Estocástico da Geometria e Heterogeneidades de Reservatórios Petrolíferos Usando o Método

Booleano.. In: I Workshop de Simulação e Engenharia de Reservatórios, 1995, Unicamp. I Workshop de

Simulação e Engenharia de Reservatórios, 1995.

Tavares, G. ; SOMBRA, C. L. ; BARRETO, A. B. ; ROMEU, R. K. ; LANZARINI, W. ; LOPES, Hélio ; PESCO, S.

. Modelagem Estocástica Booleana da Cimentação Carbonática no Arenito de Namorado do Campo de

Albacora . In: II Workshop sobre Caracterização e Engenharia de Reservatórios, 1997, Campinas - SP. II

Workshop sobre Caracterização e Engenharia de Reservatórios, 1997.

Tavares, G. ; LANZARINI, W. ; BARRETO, A. B. ; SANTOS, M. ; LOPES, Hélio ; PESCO, S. . Parametrização e

Modelagem Geométrica de Unidades Genéticas de Sistemas Deposicionais Turbidíticos. In: I Simpósio

sobre Turbiditos na Petrobras, 1997. I Simpósio sobre Turbiditos na Petrobras, 1997.

Tavares, G. ; LANZARINI, W. ; POLLETO, A. ; PESCO, S. ; LOPES, Hélio . Stochastic Modeling of Geometric

Objects and Reservoir Heterogeneities. In: Engineering Conference Exhibition Society of Petroleum

Engineers, 1997, Rio de Janeiro. Fifth Latin American and Caribbean Petroleum Engineering Conference

and Exhibition. Richardson, Texas, E. Unidos : Society of Petroleum Engineers, 1997.

Tavares, G. ; SOMBRA, C. L. ; BARRETO, A. B. ; ROMEU, R. K. ; LOPES, Hélio ; PESCO, S. . Stochastic

Modeling of Calcite Concretions in Cretaceous Turbiditic Sandstones of the Albacora Oil-Field, Campos

Basin, Brazil. In: AAPG International Conference & Exhibition, 1998, Rio de Janeiro. AAPG International

Conference & Exhibition, 1998. p. 160-161.

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References

Petbool

Tavares, G. ; LANZARINI, W. ; SANTOS, M. B. . Modelagem Estocástica Baseada em Objetos de Reservatórios

Turbidíticos do Campos da Fazenda de Queimados. In: II Seminário de Reservas e Reservatórios, 1998,

Rio de Janeiro, 1998.

Tavares, G. ; POLLETO, A. ; LANZARINI, W. . Object Based Modeling and Stochastic Simulation of Fluvial

Reservoirs Geometry. In: IAMG 99 Trondheim, 1999, Rio de Janeiro. Fifth Annual Conference of

International Association for Mathematica Geology. Trondheim : Tapir, 1999. v. 2. p. 647-652.

Tavares, G. ; LOPES, H. ; PESCO, S. ; POLETTO, C. A. . Petbool: A Software for Stochastic Modeling of

Geological Objects . In: Annual Conference of the International Association for Mathematical Geology,

2002, Berlim. Annual Conference of the International Association for Mathematical Geology, 2002.

Tavares, G. ; LOPES, H. ; PESCO, S. ; POLETTO, C. A. ; MENESES, L. ; LIMA FILHO, F. P. . Fluvial Outcrops

Parametrization Applyed to Object Based Geological Modeling for reservoirs of Potiguar Basin - Brazil . In:

Annual Conference of the International Association for Mathematical Geology, 2002, Berlim. Annual

Conference of the International Association for Mathematical Geology, 2002.

Tavares, G. ; SANTOS, Rogerio ; LOPES, Hélio ; LEWINER, Thomas ; VIEIRA, Antonio Wilson . Topological

Reconstruction of Oil Reservoirs from Seismic Surfaces. In: Annual Conference the International

Association for Mathematical Geology, 2003, Portsmouth. Conference Proceedings of the IAMG 2003,

2003. v. 1.

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References

http://www.matmidia.org

Implicit Modeling

Tavares, G. ; MIRANDA, J. . Métodos Simpliciais em Computação Gráfica. Rio de Janeiro: IMPA/CNPq, 1989.

Tavares, G. ; CASTELO, A. ; FREITAS, S. . PL Approximations of Manifolds and Application to Implicit ODEs. A.

M. S. Lecture Notes in Applied Mathematics, Estados Unidos, v. 26, p. 99-112, 1990.

Antonio Castelo. Adaptatividade de Triangulações e Aplicações à Modelagem Geométrica e Equações

Algébrico-Diferenciais. 1992. 0 f. Tese (Doutorado em Matemática) - Pontifícia Universidade Católica

do Rio de Janeiro.

Sergio Roberto Freitas. Aproximações Simpliciais de Variedades Implícitas e Equações Algébrico-Diferenciais.

1991. 0 f. Tese (Doutorado em Matemática) - Pontifícia Universidade Católica do Rio de Janeiro.

Lewiner, Thomas; Lpes, Hélio; Tavares, G. ; Vieira, Antonio Wilson . Efficient Implementation of Marching Cube

Cases with Topological Guarantees. Journal of Graphics Tools, Estados Unidos, v. 8, n. 2, p. 1-15,

2003.

CASTELO, A ; NONATO, L ; SIQUEIRA, M ; MINGHIM, R ; TAVARES, G . The J1a triangulation: An adaptive

triangulation in any dimension. Computers & Graphics, v. 30, p. 737-753, 2006.

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References

Geosis

Tavares, G. ; SANTOS, Rogerio ; LOPES, Hélio ; LEWINER, Thomas ; VIEIRA, Antonio Wilson .

Topological Reconstruction of Oil Reservoirs from Seismic Surfaces. In: Annual Conference

the International Association for Mathematical Geology, 2003, Portsmouth. Conference

Proceedings of the IAMG 2003, 2003. v. 1.

LEWINER, Thomas ; LOPES, H. ; Tavares, G. ; BORDIGNON, A. L. ; CASTRO, R. P. ;

SANTOS, Rogerio ; MURAD, A. . Reconstrução topológica tridimensional de objetos

geológicos em bacias marítimas usando dados sísmicos 2d ou 3d: uma ferramenta

exploratória e explotatória de hidrocarbonetos. In: 10th International Congress of the

Brazilian Geophysical Society, 2007, Rio de Janeiro. 10th International Congress of the

Brazilian Geophysical Society. Rio de Janeiro : Brazilian Geophysical Society, 2007.

LEWINER, Thomas ; BORDIGNON, A. L. ; CASTRO, R. P. ; LOPES, H. ; Tavares, G. ;

SANTOS, Rogerio ; MURAD, A. . Reservoir reconstruction from seismic data with

topological guarantees. In: Gocad Meeting, 2007, Vandoeuvre, France. 27th. Gocad

Meeting. Vandoeuvre, France : Nancy School of Geology, 2007.

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References

Machine Learning

Vieira, Thales ; Bordignon, Alex ; Peixoto, Adelailson ; Tavares, G. ; Lopes, Hélio ; Velho, Luiz ;

Lewiner, Thomas . Learning good views through intelligent galleries. Computer Graphics

Forum, v. 28, p. 717-726, 2009.

LAGE, M. ; CARMO, F. P. ; PAIVA NETO, A. ; LEWINER, Thomas ; Tavares, G. . Vector Field

Reconstruction from Sparse Samples with Applications. In: Sibgrapi, 2006, Manus. Sibgrapi

Proceedings, 2006.

Marcos Lage, Rener Castro, Fabiano Petronetto, Alex Bordignon, Geovan Tavares, Thomas

Lewiner, Helio Lopes. Support Vectors Learning for Vector Field Reconstrution. Sibgrapi,

2009.

João Paixão, Marcos Lage, Fabiano Petronetto, Alex Bordignon, Sinesio Pesco, Geovan

Tavares, Thomas Lewiner, Helio Lopes. Random Walks for Vector Field Denoising.

Sibgrepi, 2009.

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References

http://www.matmidia.org

KernelSis

BORDIGNON, A. L. ; CASTRO, R. P. ; LOPES, H. ; LEWINER, Thomas ; Tavares, G. ;

SANTOS, Rogerio ; MURAD, A. . Data inspection on multiatribute seismic modeling

using star coordinates. In: Gocad Meeting, 2007, Vandoeuvre, France. 27th. Gocad

Meeting. Vandoeuvre, France : School of Geology.

CASTRO, R. P. ; BORDIGNON, A. L. ; LOPES, H. ; LEWINER, Thomas ; Tavares, G. ;

SANTOS, Rogerio ; MURAD, A. . Multi-attribute seismic cell reservoir/non-reservoir

classification with kernel based methods. In: Gocad Meeting, 2007, Vandoeuvre, France.

27th. Gocad Meeting. Vandoeuvre, France : School of Geology, 2007.

• LOPES, H. ; BORDIGNON, A. L. ; CASTRO, R. P. ; LEWINER, Thomas ; Tavares, G. ;

SANTOS, Rogerio ; MURAD, A. . Mineração visual de multiatributos sísmicos para

classificação de multifácies. In: 10th International Congress of the Brazilian Geophysical

Society, 2007, Rio de Janeiro. 10th International Congress of the Brazilian Geophysical

Society. Rio de Janeiro : Brazilian Geophysical Society, 2007.

Tavares, G. ; LOPES, H. ; LEWINER, Thomas ; CASTRO, R. P. ; BORDIGNON, A. L. ; LAGE,

M. ; SANTOS, Rogerio ; MURAD, A. . Métodos baseados em núcleos e máquinas de

suporte vetorial em aplicações de geofísica de petróleo. In: 10th International Congress

of the Brazilian Geophysical Society, 2007, Rio de Janeiro. 10th International Congress

of the Brazilian Geophysical Society. Rio de Janeiro : Brazilian Geophysical Society,

2007.

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References

http://www.matmidia.org

Meshless Fluid Dynamics

Lucas Mauricio Sigaud. Estudos de Dinâmica de materiais granulares densos. 2009. Tese (Doutorado em

Física) - Pontifícia Universidade Católica do Rio de Janeiro, Conselho Nacional de Desenvolvimento

Científico e Tecnológico.

Fabiano Petronetto de Carmo. A equação de Poisson e a decomposição de Helmholtz-Hodge com operadores

SPH. 2008. Tese (Doutorado em Matemática) - Pontifícia Universidade Católica do Rio de Janeiro.

Afonso Paiva Neto. Uma abordagem lagrangeana para simulação de escoamentos de fluidos viscoplásticos e

multifásicos. 2007. Tese (Doutorado em Matemática) - Pontifícia Universidade Católica do Rio de Janeiro,

Conselho Nacional de Desenvolvimento Científico e Tecnológico.

Bordignon, A.L. ; Sigaud, L. ; Tavares, G. ; LOPES, H. ; Lewiner, T. ; Morgado, W.A.M. . Arch generated shear

bands in granular systems. Physica. A, v. 388, p. 2099-2108, 2009.

Paiva, Afonso ; Petronetto, Fabiano ; LEWINER, Thomas ; Tavares, Geovan . Particle-based viscoplastic

fluid/solid simulation. Computer Aided Design, v. 41, p. 306-314, 2009.

Petronetto, Fabiano ; Paiva, Afonso ; LAGE, M. ; LEWINER, Thomas ; Tavares, Geovan ; LOPES, H. .

Meshless Helmholtz-Hodge Decomposition.. IEEE Transactions on Visualization and Computer Graphics,

2009.

PAIVA NETO, A. ; CARMO, F. P. ; TAVARES, G ; Lewiner, T. . Simulação de Fluidos sem Malha: Uma

Introdução ao Método SPH. 1. ed. Rio de Janeiro: Instituto Nacional de Matemática Pura e Aplicada, 2009.

v. 1. 198 p.

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