Integral transform methodology for convection-diffusion problems in petroleum reservoir engineering

Almeida, Alcino R.; Cotta, Renato M.

doi:10.1016/0017-9310(95)00101-e

Cited by 31 publications

(23 citation statements)

References 5 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…This methodology uses the General Integral Transform Technique [13][14][15] to handle the nonhomogeneities of the proposed problem, by defining an eigenfunction expansion in terms of a simpler Sturm-Liouville problem of known solution 11-12-22-23 . This methodology can be combined with several analytical methodologies available in the literature 11,12,[17][18][19][20][21] to obtain different solutions for different geometries and different orthogonal coordinate systems. This methodology can be combined with several analytical methodologies available in the literature 11,12,[17][18][19][20][21] to obtain different solutions for different geometries and different orthogonal coordinate systems.…”

Section: Resultsmentioning

confidence: 99%

“…1 is the same as the one presented by Almeida and Cotta (1995) 21 for steady-state. Further cases will increase the mathematical development complexity.…”

Section: Test Case 1: Five-spot Production/injection Patern In a Homomentioning

confidence: 99%

“…The GITT has been applied to obtain analytical solution of eigenvalue problems for heterogeneous media 16 , irregular domains 17,18 , atmospheric pollutant dispersion 19 , conjugated conduction-convection heat transfer 20 , convection-diffusion problems in petroleum reservoir engineering 21 and several other non-homogeneous diffusion problems. Therefore, GITT is a robust technique that can be applied to obtain analytical solutions for diffusion problems related to reservoir engineering.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A General Analytical Solution for the Multidimensional Transient Linear Hydraulic Diffusivity Equation in Heterogeneous and Anisotropic Porous Media

Couto

Moreira²,

Marsili

2011

All Days

View full text Add to dashboard Cite

A general analytical solution for the multidimensional transient linear hydraulic diffusivity equation is presented and its application is discussed in this work. The analytical solution is valid for heterogeneous and anisotropic porous media in any orthogonal coordinate system within a regular boundary domain. A source term, dependent on position and time, was included in the partial differential equation to represent production and/or injection wells. A general boundary condition is considered to account for a constant pressure boundary, a no-flow boundary (symmetry or impermeable boundary) or any prescribed flow boundary. The Generalized Integral Transform Technique (GITT) was used to obtain the analytical solution of the proposed problem. This technique transforms a complex partial differential equation into a system of simpler ordinary differential equations that can be solved by analytical, numerical or hybrid methods. Here, we consider a fully analytical solution to the problem. The general solution is then applied to a one-phase tank model in a homogeneous and isotropic porous medium for validation and analysis. A wide variety of applications can be envisioned and discussed for the proposed general solution: well testing in regular and irregular domains, analysis of local reservoir heterogeneities, analysis of reservoir anisotropy degree, analysis of stream lines in naturally fractured and hydraulic fractured reservoirs, formation damage quantification, reservoir shape factor calculations, oil recovery in different waterflood patterns, amongst many others. These applications are discussed and analyzed in the light of the presented general analytical solution.

show abstract

Section: Resultsmentioning

confidence: 99%

“…1 is the same as the one presented by Almeida and Cotta (1995) 21 for steady-state. Further cases will increase the mathematical development complexity.…”

Section: Test Case 1: Five-spot Production/injection Patern In a Homomentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

A General Analytical Solution for the Multidimensional Transient Linear Hydraulic Diffusivity Equation in Heterogeneous and Anisotropic Porous Media

Couto

Moreira²,

Marsili

2011

All Days

View full text Add to dashboard Cite

show abstract

“…An algebraic solution of this kind has been applied to three-dimensional steady-state laminar forced convection inside rectangular ducts after integral transforming along two spatial coordinates, leaving the remaining one as the independent variable of the transformed ODE system [3]. Later on, Almeida and Cotta [1] applied the same approach to solve a transient twodimensional advection-diffusion problem in petroleum reservoirs. Quite recently, Wortmann et al [13] solved a two-dimensional steady-state advection-diffusion equation by also applying the GITT methodology, followed by a Laplace transform to obtain the analytical solution of the resulting ODE system.…”

Section: Problem Formulationmentioning

confidence: 97%

“…It is based on expansions of the original potentials in terms of eigenfunctions and the solution is obtained through integral transformation in all but one of the independent variables, thus reducing the partial differential formulations to an ordinary differential system for the expansion coefficients, which can be then solved using numerical techniques or in some special cases, analytical procedures [1,4,7].…”

Section: Introductionmentioning

confidence: 99%

Integral transform solutions for atmospheric pollutant dispersion

Almeida

Pimentel

Cotta

2007

Environ Model Assess

View full text Add to dashboard Cite

A transient two-dimensional advection-diffusion model describing the turbulent dispersion of pollutants in the atmosphere has been solved via the Generalized Integral Transform Technique (GITT), by two different schemes. The first approach performs numerical integration of the transformed system using available routines for initial value problems with automatic error control. In spite of the timeconsuming character of such a scheme, its flexibility allows the handling of problems involving time-dependent meteorological parameters such as wind speed and eddy diffusivities. The second approach works fully analytically being thus intrinsically more robust and economic, although not directly applicable in dealing with time-dependent parameters. For the test problem used in this work, both methods agree very well with each other, as well as with a known analytical solution for a simpler formulation used as benchmark. The impact of the longitudinal diffusivity on the stiffness of the ordinary differential equation (ODE) system arising from the integral transformation has been assessed through the processing time demanded to solve it when the numerical approach is used. The observed CPU times show that the analytical approach is clearly preferable unless the problem involves time-dependent parameters.

show abstract

Developing semianalytical solutions for multispecies transport coupled with a sequential first‐order reaction network under variable flow velocities and dispersion coefficients

Suk

2013

Water Resources Research

View full text Add to dashboard Cite

[1] This paper presents a semianalytical method to solve the multispecies reactive solute transport equation coupled with a sequential first-order reaction network under spatially or temporally varying flow velocities and dispersion coefficients. This method employs the generalized integral transform technique and general linear transformation method by Clement (2001) to transform the set of coupled multispecies reactive transport equations into a set of independent uncoupled equations and to solve these independent equations for spatially or temporally varying flow velocities and dispersion coefficients, as well as for a temporally varying inlet concentration. The proposed semianalytical solution is compared against previously published analytical solutions of Clement (2008b) and van Genuchten (1985). An example is used to show application of the solution to a hypothetical multilayered medium. The solution of proposed approach is also compared with a numerical solution using the 2DFATMIC (Two-Dimensional Subsurface Flow, Fate and Transport of Microbes and Chemicals Model). Three scenarios are illustrated to show the capabilities of the proposed semianalytical method to deal with aquifer heterogeneity and transient situations. We also show a practical implementation of the solution to an actual field, single-well push-pull test example designed to obtain the concentration distribution of reactants consumed and products formed at the end of the injection phase.

show abstract

Integral transform methodology for convection-diffusion problems in petroleum reservoir engineering

Cited by 31 publications

References 5 publications

A General Analytical Solution for the Multidimensional Transient Linear Hydraulic Diffusivity Equation in Heterogeneous and Anisotropic Porous Media

A General Analytical Solution for the Multidimensional Transient Linear Hydraulic Diffusivity Equation in Heterogeneous and Anisotropic Porous Media

Integral transform solutions for atmospheric pollutant dispersion

Developing semianalytical solutions for multispecies transport coupled with a sequential first‐order reaction network under variable flow velocities and dispersion coefficients

Contact Info

Product

Resources

About