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Research Papers: Energy Systems Analysis

Orthogonal Collocation Method for Solving the Diffusivity Equation: Application on Dual Porosity Reservoirs With Constant Pressure Outer Boundary

[+] Author and Article Information
Mahshid Nategh

Department of Chemical Engineering,
School of Chemical and Petroleum Engineering,
Shiraz University,
Shiraz 71964-84334, Iran

Behzad Vaferi

Young Researchers and Elite Club,
Shiraz Branch,
Islamic Azad University,
Shiraz 74731-71987, Iran
e-mails: behzad.vaferi@gmail.com,
vaferi@iaushiraz.ac.ir

Masoud Riazi

Enhanced Oil Recovery (EOR) Research Centre,
IOR/EOR Research Institute,
Shiraz University, Shiraz 71964-84334, Iran

1Corresponding author.

Contributed by the Advanced Energy Systems Division of ASME for publication in the JOURNAL OF ENERGY RESOURCES TECHNOLOGY. Manuscript received June 22, 2018; final manuscript received October 19, 2018; published online November 19, 2018. Assoc. Editor: Reza Sheikhi.

J. Energy Resour. Technol 141(4), 042001 (Nov 19, 2018) (9 pages) Paper No: JERT-18-1452; doi: 10.1115/1.4041842 History: Received June 22, 2018; Revised October 19, 2018

Fluid flow inside heterogeneous structure of dual porosity reservoirs is presented by two coupled partial differential equations (PDE). Finding an analytical solution for the diffusivity equations is tedious or even impossible in some circumstances due to the heterogeneity of dual porosity reservoirs. Therefore, in this study, orthogonal collocation method (OCM) is proposed for solving the governing equations in dual porosity reservoirs with constant pressure outer boundary. Since no analytical solution has been proposed for this system, validation is carried out by comparing the OCM-obtained results for “dual porosity reservoirs with circular no-flow outer boundary” with both exact analytical solution and real field data. Sensitivity analyses reveal that the OCM with 13 collocation points is a good candidate for prediction of pressure transient response (PTR) in dual porosity reservoirs. OCM predicts the PTR of a real field draw-down test with an absolute average relative deviation (AARD) of 0.9%. Moreover, OCM shows a good agreement with the analytical solution obtained by Laplace transform (AARD = 0.16%). It is worth noting that OCM requires a smaller computational effort. Thereafter, PTR of dual porosity reservoirs with a constant production rate in the wellbore and constant pressure outer boundary is simulated by OCM for wide ranges of operating conditions. Accuracy of OCM and its low required computational time justifies that this approximate method can be considered as a practical candidate for pressure transient analysis in dual porosity reservoirs.

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Figures

Grahic Jump Location
Fig. 1

Schematic of realistic (a) and conceptual grid model (b) for the fractured reservoirs [34]

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Fig. 2

Schematic of the considered dual porosity reservoir, constant rate production from wellbore, and constant pressure outer boundary

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Fig. 3

Predicted pressure by the OCM and the real field pressure data for the Mach-3X well [47]

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Fig. 4

Dimensionless pressure as a function of dimensionless time

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Fig. 5

Dimensionless pressure as a function of dimensionless time for different radial distance from the wellbore

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Fig. 6

Pressure derivative graph obtained from the simulated PTR by orthogonal collocation

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Fig. 7

Dimensionless pressure derivative graph as a function of dimensionless time for different values of ω

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Fig. 8

Dimensionless PD graph as a function of dimensionless time for different values of λ

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Fig. 9

Dimensionless pressure profile in the dual porosity reservoir as a function of dimensionless time and radial distance from the wellbore

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