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Numerical Modeling and Spatial Moment Analysis of Solute Mobility and Spreading in a Coupled Fracture-Skin-Matrix System

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Abstract

This study deals with transport of solutes through a saturated sub-surface rock formation with well-defined horizontal parallel fractures. For this purpose, a simplified conceptual model consisting of a single fracture and its associated rock-matrix is considered in the presence of a fracture-skin in order to study the mobility and mixing of solutes along the fracture. In this paper, a coupled fracture-skin-matrix system is modeled numerically using finite difference method in a pseudo two-dimensional domain with a constant continuous source at fracture inlet. Flow and transport processes are considered parallel to the fracture axis, while the transport processes in fracture-skin as well as in rock-matrix are considered perpendicular to the fracture axis. Having obtained the concentration distribution along the fracture, method of spatial moments is employed to study the mobility and spreading of solutes. Sensitivity analyses have been done to understand the effect of various fracture-skin parameters like porosity, thickness, and diffusion coefficient. Further, the influence of non-linear sorption and radioactive decaying of solutes are carried out for different sorption intensities and decay constants. Results suggest that the presence of fracture-skin significantly influences the mobility and spreading of solutes along the fracture in comparison with a coupled fracture-matrix system without fracture-skin.

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Abbreviations

b:

Half fracture aperture, (L)

d:

Distance from center of fracture to the skin-matrix interface, (L)

H:

Half fracture spacing, (L)

Cf :

Volume concentration of solute in the fracture, (ML−3)

Cs :

Volume concentration of solute in the skin, (ML−3)

Cm :

Volume concentration of solute in the matrix, (ML−3)

\( C_{s\quad 1}^{n + 1} \) :

Volume concentration of the solute at the first node in the fracture skin, (ML−3)

\( C_{s\quad 2}^{n + 1} \) :

Volume concentration of the solute at the second node in the fracture skin, (ML−3)

C:

Volume concentration of solute, (ML−3)

C0 :

Constant concentration for continuous injection of solute source at the inlet of fracture, (ML−3)

DL :

Hydrodynamic dispersion coefficient in the fracture, (L2T−1)

D* :

Free molecular diffusion coefficient of the solute in water, (L2T−1)

DS :

Effective diffusion coefficient in the fracture-skin, (L2T−1)

Dm :

Effective diffusion coefficient in the rock-matrix, (L2T−1)

Kd :

Proportion or distribution coefficient, ((L3M−1)n)

Kf :

Surface sorption coefficient of fracture, (L)

KS :

Volume sorption coefficient of the fracture-skin, (L3M−1)

Km :

Volume sorption coefficient of the rock-matrix, (L3M−1)

Lf :

Length of the fracture along the flow direction, (L)

n:

Freundlich sorption isotherm exponent, (−)

Rf :

Retardation factor of fracture, (−)

RS :

Retardation factor of fracture-skin, (−)

Rm :

Retardation factor of rock-matrix, (−)

S:

Sorbed concentration, (MM−1)

λ:

Radioactive decay constant, (T−1)

t:

Time variable, (T)

V0 :

Mean water velocity in the fracture, (LT−1)

V(t):

Velocity of the solute in the fracture corresponding to time t, (LT−1)

D(t):

Macro-dispersion coefficient in the fracture corresponding to time t, (LT−1)

x:

Space coordinate along the flow direction in the fracture plane, (L)

y:

Space coordinate in the direction normal to the fracture, (L)

\( \bar{x} \) :

Mean distance traveled by a solute front along the fracture, (L)

α0 :

Local fracture dispersivity, (L)

ρ s :

Bulk density of the fracture skin, (ML−3)

θ s :

Fracture-skin porosity, (−)

ρ m :

Bulk density of the rock matrix, (ML−3)

θ m :

Rock-matrix porosity, (−)

Δy (1):

Cell width across the fracture-skin interface, (L)

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Acknowledgments

The financial support by Indian Institute of Technology, Madras in New Faculty Scheme with Project No: CIE/07-08/202/NFSC/GSUR is gratefully acknowledged.

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Authors and Affiliations

  1. EWRE Division, Department of Civil Engineering, IIT-Madras, Chennai, 600036, India

    V. Renu

  2. Petroleum Engineering Program, Department of Ocean Engineering, IIT-Madras, Chennai, 600036, India

    G. Suresh Kumar

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  1. V. Renu
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Correspondence to G. Suresh Kumar.

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Renu, V., Suresh Kumar, G. Numerical Modeling and Spatial Moment Analysis of Solute Mobility and Spreading in a Coupled Fracture-Skin-Matrix System. Geotech Geol Eng 30, 1289–1302 (2012). https://doi.org/10.1007/s10706-012-9540-3

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