9/8/2006 1 Trefftz Trefftz - - Herrera Collocation Herrera Collocation Method for Combustion Fronts in Method for Combustion Fronts in Oil Reservoirs: Application Oil Reservoirs: Application M. A. Díaz M. A. Díaz - - Viera, D. A. López Viera, D. A. López - - Falcón Falcón Instituto Mexicano del Petróleo, Instituto Mexicano del Petróleo, I. Herrera I. Herrera Instituto de Geof Instituto de Geof í í sica, UNAM sica, UNAM 7th World 7th World Congress Congress on on Computational Computational Mechanics Mechanics Hyatt Hyatt Regency Regency Century Century Plaza Hotel Plaza Hotel Los Los Angeles Angeles , California , California July July 16 16 - - 22, 2006 22, 2006
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Trefftz-Herrera Collocation Method for Combustion Fronts in Oil
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9/8/2006 1
TrefftzTrefftz--Herrera Collocation Herrera Collocation Method for Combustion Fronts in Method for Combustion Fronts in
M. A. DíazM. A. Díaz--Viera, D. A. LópezViera, D. A. López--Falcón Falcón Instituto Mexicano del Petróleo, Instituto Mexicano del Petróleo,
I. HerreraI. HerreraInstituto de GeofInstituto de Geofíísica, UNAMsica, UNAM
7th World 7th World CongressCongress onon ComputationalComputational MechanicsMechanicsHyattHyatt RegencyRegency CenturyCentury Plaza HotelPlaza Hotel
Los Los AngelesAngeles, California, CaliforniaJulyJuly 16 16 -- 22, 200622, 2006
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IntroductionIntroduction
Recently, some authors (H. Recently, some authors (H. HoteitHoteit andand FiroozabadiFiroozabadi, 2005) , 2005) in the specialized oil reservoir literature have suggested the in the specialized oil reservoir literature have suggested the application of Discontinuous application of Discontinuous GalerkinGalerkin Method to capture Method to capture sharp moving fronts in modeling sharp moving fronts in modeling multicomponentmulticomponent fluid fluid flow in fractured mediaflow in fractured media.
In the present work we will consider, as a very competitive In the present work we will consider, as a very competitive alternative method, the alternative method, the TrefftzTrefftz--Herrera approach in Herrera approach in conjunction with collocation to obtain the numerical conjunction with collocation to obtain the numerical solution of a system of boundary value problems with solution of a system of boundary value problems with prescribed jumps (prescribed jumps (BVPJ’sBVPJ’s) which models the dynamics of ) which models the dynamics of combustion fronts in porous media.combustion fronts in porous media.
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IntroductionIntroduction
TrefftzTrefftz--Herrera Collocation Method was introduced in Herrera Collocation Method was introduced in previous papers: previous papers:
–– Herrera, I.; Diaz, M. “Indirect Methods of Collocation: Herrera, I.; Diaz, M. “Indirect Methods of Collocation: TrefftzTrefftz--Herrera Herrera Collocation”. Numerical Methods for Partial Differential EquatioCollocation”. Numerical Methods for Partial Differential Equations. ns. 1515(6) 709(6) 709--738, 1999.738, 1999.
–– Herrera, I., Yates R. and Diaz M. “General Theory of Domain Herrera, I., Yates R. and Diaz M. “General Theory of Domain Decomposition: Indirect Methods”. Numerical Methods for Partial Decomposition: Indirect Methods”. Numerical Methods for Partial Differential Equations, Vol. 18, No. 3, pp. 296Differential Equations, Vol. 18, No. 3, pp. 296--322, may. 2002.322, may. 2002.
–– Diaz M. and I. Herrera, “THDiaz M. and I. Herrera, “TH--Collocation for the Biharmonic Collocation for the Biharmonic Equation”, Equation”, Advances in Engineering SoftwareAdvances in Engineering Software,, Volume 36, Issue 4, Volume 36, Issue 4, Pages 243Pages 243--251, April 2005.251, April 2005.
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IntroductionIntroduction
The distinguishing feature of the The distinguishing feature of the TrefftzTrefftz--Herrera Herrera procedure is the use of specialized test functions, that procedure is the use of specialized test functions, that could be in general discontinuous, becomes it in a very could be in general discontinuous, becomes it in a very attractive choice for solving problems with prescribed attractive choice for solving problems with prescribed jumps and with discontinuous coefficients.jumps and with discontinuous coefficients.
In particular, the numerical modeling of a planar In particular, the numerical modeling of a planar combustion front for the steady state under adiabatic combustion front for the steady state under adiabatic conditions is implemented. Some preliminary numerical conditions is implemented. Some preliminary numerical results and its comparison with the analytical solutions results and its comparison with the analytical solutions are presented.are presented.
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Collocation MethodsCollocation Methods
Collocation is known as an efficient and highly accurate Collocation is known as an efficient and highly accurate numerical solution procedure for partial differential equations.numerical solution procedure for partial differential equations.
Usually this kind of methods are applied using Usually this kind of methods are applied using splinessplines. However, . However, a more general point of view is obtained when they are a more general point of view is obtained when they are formulated using the approach proposed by Herrera, in spaces formulated using the approach proposed by Herrera, in spaces of fully discontinuous functions; i.e., spaces in which the of fully discontinuous functions; i.e., spaces in which the functions and their derivatives may have jump discontinuities. functions and their derivatives may have jump discontinuities.
In the case of elliptic equations of second order, it is standarIn the case of elliptic equations of second order, it is standard d requiring continuity of both, the function and its derivative. requiring continuity of both, the function and its derivative. However, these conditions can be relaxed when However, these conditions can be relaxed when TrefftzTrefftz--Herrera Herrera method is applied. method is applied.
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Classification of Collocation Classification of Collocation MethodsMethods
Direct and indirect methodsDirect and indirect methods: Direct methods are those in : Direct methods are those in which collocation is used to construct the solution directly, whwhich collocation is used to construct the solution directly, while ile indirect methods are those in which collocation is applied to indirect methods are those in which collocation is applied to construct construct specialized test functionsspecialized test functions..
Overlapping and nonOverlapping and non--overlapping methodsoverlapping methods: depending on : depending on whether the whether the subregionssubregions used in the construction of the solution used in the construction of the solution are disjoint or overlapping.are disjoint or overlapping.
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Diagram of Collocation MethodsDiagram of Collocation Methods
ConclusionsConclusionsHere, we have presented some preliminary numerical results that, as was expected, have shown a good agreement with the analytical solutions.
In this sense, TH-Collocation method has proved to be a very efficient framework to obtain highly accurate numerical solutions of the BVPJ’s which describe the dynamics of a planar combustion front for the steady state case under adiabatic conditions.
This is only the starting point in the numerical simulation of the more general case which is a subject of future work.
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Future WorkFuture WorkWe intend to consider the following cases:
Non Adiabatic ConditionsNon Linear problems (non constant gas phase density)Transient problems. 2D and 3D problems (reservoir scale)More general front geometries.
Combustion front tracking: Local Grid Refinement Methods or Level Set Methods (Osher & Sethian)
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!!!Thank you, for you attention!!!Thank you, for you attentionWe are open to hear We are open to hear