Solute transport in a single fracture with negligible matrix permeability: 2. mathematical formalism

Abstract

This report provides an overview of the mathematical expressions for modeling fundamental solute transport mechanisms at the fracture scale. It focuses on low-permeability rocks where advection in the matrix is negligible as compared to that in fractures. The following processes are considered: (1) advective transport in fractures, (2) hydrodynamic dispersion along the fracture axis, (3) molecular diffusion from the fracture to the porous matrix, (4) sorption reactions on the fracture walls and within the matrix, and (5) decay reactions. The aim of this review is to gather in a single article the transport equations and their analytical solutions, using a homogeneous notation to facilitate comparison and exploitation.

Résumé

Ce rapport présente une vue d'ensemble des expressions mathématiques proposées dans la littérature pour modéliser le transport de soluté à l'échelle de la fracture. L'accent est mis sur les roches faiblement perméables où le transport par convection dans la matrice rocheuse peut être considéré comme négligeable devant le transport dans les fractures. Les processus considérés sont les suivants: (1) transport par convection dans les fractures, (2) dispersion hydrodynamique, (3) échanges de soluté par diffusion moléculaire entre les fractures et la matrice rocheuse, (4) réactions de sorption sur les parois des fractures et à l'intérieur de la matrice et (5) réactions de décroissance. L'objectif premier de cette revue est de rassembler les équations de transport et leurs solutions analytiques en utilisant une notation homogène pour faciliter leur utilisation et les comparaisons.

Resumen

Este informe proporciona una revisión de las expresiones matemáticas para modelar los mecanismos fundamentales de transporte de solutos a escala de fracturas. Se centra en rocas de baja permeabilidad donde la advección en la matriz es despreciable en comparación con la de las fracturas. Se considera los procesos siguientes: (1) transporte advectivo en las fracturas; (2) dispersión hidrodinámica a lo largo del eje de la fractura; (3) difusión molecular desde la fractura a la matriz porosa; (4) reacciones de sorción en la pared de la fractura y en la matriz; y (5) reacciones de degradación. El propósito de esta revisión es reunir en un único artículo las ecuaciones de transporte y sus soluciones analíticas con una notación homogénea para facilitar su comparación y aplicación.

Notation

c _f [M·L ^–3]:

Volume concentration of solute in the fracture

t [T]:

Time variable

x [L]:

Space coordinate along the flow direction in the fracture plane

u [L·T ^–1]:

Mean fluid velocity in the fracture

u _i [L·T ^–1]:

Scalar component of the fluid velocity vector along the i direction (i = x, y, z)

D _L [L ²·T ^–1]:

Hydrodynamic dispersion coefficient in the fracture

α _L [L]:

Dispersivity in the fracture

D _m [L ²·T ^–1]:

Molecular-diffusion coefficient of the solute in free water

Re [−]:

Reynolds number

ε [L]:

Fracture roughness (mean height of the fracture asperity)

Rr [−]:

Relative roughness

b [L]:

Half aperture of the fracture

W [L]:

Width of the fracture in the direction perpendicular to the pressure gradient

υ [L ²·T ^–1]:

Kinematic viscosity

μ [M·L ^–1·T ^–1]:

Dynamic viscosity

D _h [L]:

Hydraulic diameter of the fracture

S [L ²]:

Flow section of the fracture

P _ext [L]:

External perimeter of the flow section

ρ [M·L ^–3]:

Mass density of the fluid

g _i [L ²·T ^–1]:

Scalar component of the gravitational acceleration vector along the i direction (i=x, y, z)

Q _{f unit} [L ²·T ^–1]:

Flow rate per width unit in the fracture

Q _f [L ³·T ^–1]:

Total flow rate in the fracture

P [M·L ^–1·T ²]:

Fluid pressure

H [L]:

Hydraulic head

K _f [L·T ^–1]:

Hydraulic conductivity of the fracture.

m ₀ [M]:

Injected mass

c ₀ [M·L ^–3]:

Constant concentration for a continuous injection

k _B [M·L ²·T ^–2·K ^–1]:

Boltzmann constant

T [K]:

Absolute temperature

r _p [L]:

Radius of the solute molecules

τ _c [T]:

Critical time needed for a particle to experience the whole cross-sectional profile of velocities in a fracture without sorption reaction on the fracture walls

τ′ _c [T]:

Critical time needed for a particle to experience the whole cross-sectional profile of velocities in a fracture with sorption reaction on the fracture walls

x _c [L]:

Minimum travel distance for which the Taylor–Aris dispersion regime is completely established in a parallel-plate fracture

R _a [−]:

Retardation factor

〈2b〉 [L]:

Mean-fracture aperture in a fracture with variable aperture

λ _β [L]:

Correlation length of the logarithm of the apertures in a fracture with variable aperture

σ _β [−]:

Standard deviation of the logarithm of the apertures in a fracture with variable aperture

α _g [L]:

Longitudinal macro-dispersivity in a fracture with variable aperture

h [L]:

Lag distance (variogram analysis)

P _e [-]:

Peclet number

[φ] _s :

Concentration of the element φ in the solid phase, expressed as a mass per unit surface [M·L ^–2] for a sorption on the fracture walls, or as a mass per unit mass of solid [M·M ^–1] for sorption onto the matrix

[φ] _aq [M·L ^–3]:

Concentration of the element φ in the fluid phase

K _d [L] or [L ³·M ^–1]:

Sorption coefficient

a _w [L ^–1]:

Flow-wetted surface

R _f [−]:

Retardation coefficient (solute sorption on the fracture walls)

$\left[ \varphi \right]_s^0$ [M·L ^–2] or [M·M ^–1]:

Maximal concentration adsorbed onto the solid phase

k _ads [L·T ^–1] or [L ³·M ^–1·T ^–1]:

Adsorption rate

k _dsp [T ^–1]:

Desorption rate

λ [T ^–1]:

Decay constant

t ^1/2 [T]:

Half-life period

$c_f^i ,c_f^j$ [M·L ^–3]:

Aqueous concentration of elements i and j in the fracture (decay chain reaction)

$R_f^i ,R_f^j$ [−]:

Retardation coefficient of elements i and j (decay chain reaction)

ζ _ij [−]:

Fraction of parent element i transformed into element j (decay chain reaction)

M [−]:

Total number of parent elements i transformed into element j (decay chain reaction)

D _p [L ²·T ^–1]:

Pore-diffusion coefficient

D _e [L ²·T ^–1]:

Effective-diffusion coefficient

D _a [L ²·T ^–1]:

Apparent diffusion coefficient

δ _D [−]:

Pore constrictivity

τ ²[−]:

Pore tortuosity

F _f [-]:

Formation factor

c _m [M·L ^-3]:

Aqueous-solute concentration in the rock matrix

θ _m [−]:

Rock-matrix porosity

θ _t [−]:

Transport porosity of the rock-matrix

θ _s [−]:

Storage porosity of the rock-matrix

c _s [M·M ⁻¹]:

Concentration of the sorbed solute in the solid phase of the rock matrix

D _s [L ²·T ^–1]:

Diffusion coefficient in the sorbed phase

α [−]:

Capacity factor including the sorption properties of the matrix

ρ _m [M·L ^–3]:

Bulk density of the rock matrix

β _F [−]:

Freundlich exponent

A [L ²]:

Cross-sectional area of the matrix block normal to the transport direction in the fracture

B [L]:

Maximal depth of solute penetration in the matrix by diffusion

t′ ₀ [T]:

Mean residence time in the fracture

p [£]:

Laplace parameter

T* [T]:

Half-period of the Fourier series for the numerical inversion of the Laplace transform

E [M·L ^–3]:

Absolute error tolerance for the numerical inversion of the Laplace transform

ρ _c [M·L ^–3]:

Bulk density of the fracture-wall coating

δ _c [L]:

Thickness of the fracture-wall coating

c _im [M·L ^–3]:

Aqueous-solute concentration in the immobile zone (Coats–Smith model)

θ [−]:

Volumetric fraction of the mobile zone (Coats-Smith model)

θ′[−]:

Volumetric fraction of the immobile zone (Coats-Smith model)

ω [T ^–1]:

Mass-transfer coefficient (Coats-Smith model)

(c _im ) _j [M·L ^–3]:

Aqueous-solute concentration within the j ^th immobile zone (multiple-rate model)

β _j [−]:

Ratio of total-solute mass in the immobile zone j, to the mass in the mobile zone at equilibrium (multiple-rate model)

α _j [T ^–1]:

Apparent first-order mass-transfer coefficient associated with the j ^th immobile zone (multiple-rate model)