Mathematical modelling of flow towards an oil well

We describe the numerical approximations and applications of a mathematical model that governs the flow of oil towards a well. The flow of a single-phase fluid in a porous medium is governed by a parabolic equation obtained by combining the Darcy's and the continuity equations. In order to acco...

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Autores principales: Jacovkis, P.M., Savioli, G.B., Bidner, M.S.
Formato: JOUR
Materias:
Finite differences
Oil flow
Simulation
Stability
Computer simulation
Convergence of numerical methods
Equations of motion
Finite difference method
Initial value problems
Iterative methods
Linear systems
Mathematical models
Matrix algebra
Alternating direction method
Block successive over relaxation method
Darcy law
Oil wells
finite difference technique
mathematical modeling
oil hydraulics
porous medium
Acceso en línea:http://hdl.handle.net/20.500.12110/paper_00295981_v46_n9_p1521_Jacovkis
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Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) de Universidad de Buenos Aires
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spelling todo:paper_00295981_v46_n9_p1521_Jacovkis2023-10-03T14:39:27Z Mathematical modelling of flow towards an oil well Jacovkis, P.M. Savioli, G.B. Bidner, M.S. Finite differences Oil flow Simulation Stability Computer simulation Convergence of numerical methods Equations of motion Finite difference method Initial value problems Iterative methods Linear systems Mathematical models Matrix algebra Alternating direction method Block successive over relaxation method Darcy law Oil wells finite difference technique mathematical modeling oil hydraulics porous medium We describe the numerical approximations and applications of a mathematical model that governs the flow of oil towards a well. The flow of a single-phase fluid in a porous medium is governed by a parabolic equation obtained by combining the Darcy's and the continuity equations. In order to account for the spatial variations of porosity and permeability, and for permeability anisotropy, a two-dimensional model is put forward. A mixed initial-boundary value problem is numerically solved by a finite-difference family of numerical schemes, which depends on a parameter θ. The stability - conditional or unconditional, depending on θ - and the convergence of the schemes have been proved. The linear system originated at each time step is solved by the iterative ADI and block-SOR methods, and by a Taylor series of matrix functions (TSMF). These methods are compared and their relative efficiencies are carefully assessed. TSMF is the fastest technique given that adequate values of θ and time step Δt are used - but Δt must remain small. A combination of TSMF and block-SOR with variable At seems to be the best policy. Our numerical simulator is tested by reproducing the existing analytical solutions for limiting cases, and then applied in well test analysis. The contributions of this work are: (1) we introduce the TSMF technique to reservoir simulation and (2) vertical permeability and permeability spatial variations are included in a well test simulator for further developments. Copyright © 1999 John Wiley & Sons, Ltd. JOUR info:eu-repo/semantics/openAccess http://creativecommons.org/licenses/by/2.5/ar http://hdl.handle.net/20.500.12110/paper_00295981_v46_n9_p1521_Jacovkis
institution Universidad de Buenos Aires
institution_str I-28
repository_str R-134
collection Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA)
topic Finite differences
Oil flow
Simulation
Stability
Computer simulation
Convergence of numerical methods
Equations of motion
Finite difference method
Initial value problems
Iterative methods
Linear systems
Mathematical models
Matrix algebra
Alternating direction method
Block successive over relaxation method
Darcy law
Oil wells
finite difference technique
mathematical modeling
oil hydraulics
porous medium
spellingShingle Finite differences
Oil flow
Simulation
Stability
Computer simulation
Convergence of numerical methods
Equations of motion
Finite difference method
Initial value problems
Iterative methods
Linear systems
Mathematical models
Matrix algebra
Alternating direction method
Block successive over relaxation method
Darcy law
Oil wells
finite difference technique
mathematical modeling
oil hydraulics
porous medium
Jacovkis, P.M.
Savioli, G.B.
Bidner, M.S.
Mathematical modelling of flow towards an oil well
topic_facet Finite differences
Oil flow
Simulation
Stability
Computer simulation
Convergence of numerical methods
Equations of motion
Finite difference method
Initial value problems
Iterative methods
Linear systems
Mathematical models
Matrix algebra
Alternating direction method
Block successive over relaxation method
Darcy law
Oil wells
finite difference technique
mathematical modeling
oil hydraulics
porous medium
description We describe the numerical approximations and applications of a mathematical model that governs the flow of oil towards a well. The flow of a single-phase fluid in a porous medium is governed by a parabolic equation obtained by combining the Darcy's and the continuity equations. In order to account for the spatial variations of porosity and permeability, and for permeability anisotropy, a two-dimensional model is put forward. A mixed initial-boundary value problem is numerically solved by a finite-difference family of numerical schemes, which depends on a parameter θ. The stability - conditional or unconditional, depending on θ - and the convergence of the schemes have been proved. The linear system originated at each time step is solved by the iterative ADI and block-SOR methods, and by a Taylor series of matrix functions (TSMF). These methods are compared and their relative efficiencies are carefully assessed. TSMF is the fastest technique given that adequate values of θ and time step Δt are used - but Δt must remain small. A combination of TSMF and block-SOR with variable At seems to be the best policy. Our numerical simulator is tested by reproducing the existing analytical solutions for limiting cases, and then applied in well test analysis. The contributions of this work are: (1) we introduce the TSMF technique to reservoir simulation and (2) vertical permeability and permeability spatial variations are included in a well test simulator for further developments. Copyright © 1999 John Wiley & Sons, Ltd.
format JOUR
author Jacovkis, P.M.
Savioli, G.B.
Bidner, M.S.
author_facet Jacovkis, P.M.
Savioli, G.B.
Bidner, M.S.
author_sort Jacovkis, P.M.
title Mathematical modelling of flow towards an oil well
title_short Mathematical modelling of flow towards an oil well
title_full Mathematical modelling of flow towards an oil well
title_fullStr Mathematical modelling of flow towards an oil well
title_full_unstemmed Mathematical modelling of flow towards an oil well
title_sort mathematical modelling of flow towards an oil well
url http://hdl.handle.net/20.500.12110/paper_00295981_v46_n9_p1521_Jacovkis
work_keys_str_mv AT jacovkispm mathematicalmodellingofflowtowardsanoilwell
AT savioligb mathematicalmodellingofflowtowardsanoilwell
AT bidnerms mathematicalmodellingofflowtowardsanoilwell
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