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Research Papers: Petroleum Engineering

Dynamic Dispersion Coefficient of Solutes Flowing in a Circular Tube and a Tube-Bundle Model

[+] Author and Article Information
Xiaoyan Meng

Petroleum Systems Engineering,
Faculty of Engineering and Applied Science,
University of Regina,

Daoyong Yang

Petroleum Systems Engineering,
Faculty of Engineering and Applied Science,
University of Regina,
e-mail: tony.yang@uregina.ca

1Corresponding author.

Contributed by the Petroleum Division of ASME for publication in the JOURNAL OF ENERGY RESOURCES TECHNOLOGY. Manuscript received May 17, 2017; final manuscript received July 10, 2017; published online August 22, 2017. Editor: Hameed Metghalchi.

J. Energy Resour. Technol 140(1), 012903 (Aug 22, 2017) (12 pages) Paper No: JERT-17-1232; doi: 10.1115/1.4037374 History: Received May 17, 2017; Revised July 10, 2017

Abstract

Mathematical formulations have been proposed and verified to determine dynamic dispersion coefficients for solutes flowing in a circular tube with fully developed laminar flow under different source conditions. Both the moment analysis method and the Green's function are used to derive mathematical formulations, while the three-dimensional (3D) random walk particle tracking (RWPT) algorithm in a Cartesian coordinate system has been modified to describe solute flow behavior. The newly proposed formulations have been verified to determine dynamic dispersion coefficients of solutes by achieving excellent agreements with both the RWPT results and analytical solutions. The differences among transverse average concentration using the Taylor model with and without dynamic dispersion coefficient and center-of-mass velocity are significant at early times but indistinguishable when dimensionless time ($tD$) approaches 0.5. Furthermore, compared to solutes flowing in a 3D circular tube, dispersion coefficients of solutes flowing in a two-dimensional (2D) parallel-plate fracture are always larger for a uniform planar source; however, this is not always true for a point source. Solute dispersion in porous media represented by the tube-bundle model is greatly affected by pore-size distribution and increases as standard deviation of pore-size distribution ($σ$) increases across the full-time scale.

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Figures

Fig. 1

Schematics of solutes transport in a circular tube with fully developed laminar flow

Fig. 2

Snapshots of RWPT simulations for solutes originating from an instantaneous point source (x=0,r′=0) in x-, y-, and z-directions in the tube system at (a) tD=0.1 and (b) tD=1

Fig. 3

Comparisons of solute dynamic dispersion coefficients calculated using Eq. (5a) or Eq. (5b) (curves) and RWPT simulations (symbols) for instantaneous (a) point; and (b) uniform planar sources

Fig. 4

Comparisons of solute dynamic dispersion coefficients for an instantaneous uniform planar source (triangles) calculated with Eq. (5b) and analytical solutions of Aris ([17], dashed line) and Barton ([20], solid line, Eq. (10))

Fig. 5

(a) Comparisons of solute dynamic dispersion coefficient of solutes calculated from Eq. (5); and (b) the center-of-mass velocities from Eq. (6) for the three different point sources

Fig. 6

Comparisons of transverse average concentration as a function of distance using Eq. (11) with dynamic dispersion coefficient and νc, dynamic dispersion coefficient and ν¯, and Aris-Taylor dispersion coefficient and ν¯ under different source conditions at (a) tD=0.01, (b) tD=0.05, (c) tD=0.1, and (d) tD=0.5

Fig. 7

(a) Solute dynamic dispersion coefficients in a parallel-plate fracture and a circular tube for the instantaneous point source (x=0, r′=0; x=0, y′=0); and (b) ratio of solute dynamic dispersion coefficients in a parallel-plate fracture and a circular tube for different source conditions

Fig. 8

Snapshots of RWPT results for solutes released as instantaneous point sources (a) in y- and z-directions at tD=0.1 in a tube system (x=0,r′=0); (b) in y- and z-directions at tD=1 in the tube system (x=0, r′=0); (c) in x- and y-directions at tD=0.1 for a parallel-plate fracture (x=0, y′=0); and (d) in x- and y-directions at tD=0.1 for a tube system (x=0, r′=0)

Fig. 9

Pore-size histogram for Gaussian distribution at r¯=1 cm (a) σ=1 cm2, (b) σ=5 cm2, and (c) σ=10 cm2

Fig. 10

(a) Comparison of temporal evolution of solute dispersion coefficient with tube-bundle model for Gaussian pore-size distribution at different σ; and (b) decompositions of solute dispersion coefficient with tube-bundle model for Gaussian pore-size distribution at σ=5 cm2

Fig. 11

Comparison of f shown in Eq. (12a) for Gaussian pore-size distribution at different σ

Fig. 12

(a) Comparison of temporal evolution of solute dispersion coefficient with tube-bundle model for Lognormal pore-size distribution at different σ; and (b) comparison of f shown in Eq. (12a) for Lognormal and Gaussian pore-size distribution at different σ

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