Review: Mathematical expressions for estimating equivalent permeability of rock fracture networks

Abstract

Fracture networks play a more significant role in conducting fluid flow and solute transport in fractured rock masses, comparing with that of the rock matrix. Accurate estimation of the permeability of fracture networks would help researchers and engineers better assess the performance of projects associated with fluid flow in fractured rock masses. This study provides a review of previous works that have focused on the estimation of equivalent permeability of two-dimensional (2-D) discrete fracture networks (DFNs) considering the influences of geometric properties of fractured rock masses. Mathematical expressions for the effects of nine important parameters that significantly impact on the equivalent permeability of DFNs are summarized, including (1) fracture-length distribution, (2) aperture distribution, (3) fracture surface roughness, (4) fracture dead-end, (5) number of intersections, (6) hydraulic gradient, (7) boundary stress, (8) anisotropy, and (9) scale. Recent developments of 3-D fracture networks are briefly reviewed to underline the importance of utilizing 3-D models in future research.

Résumé

Les réseaux de fracture jouent un rôle plus significatif dans l’écoulement d’un fluide et du transport de soluté dans les massifs rocheux fracturés, en comparaison avec celui de la matrice rocheuse. Une estimation précise de la perméabilité des réseaux de fracture aiderait les chercheurs et les ingénieurs à mieux évaluer la performance des projets liés à l’écoulement du fluide dans les massifs rocheux fracturés. Cette étude présente un examen des travaux antérieurs qui ont porté sur l’estimation de la perméabilité équivalente de réseaux de fractures discrétisés (RFDs) en deux dimensions (2-D) compte tenu des influences des propriétés géométriques des massifs rocheux fracturés. Les expressions mathématiques pour les effets de neuf paramètres importants qui ont une incidence significative sur la perméabilité équivalente des RFDs sont résumées, comprenant (1) la distribution des longueurs de fracture, (2) la distribution des ouvertures, (3) la rugosité de la surface des fractures, (4) les fractures sans issues , (5) le nombre d’intersections, (6) le gradient hydraulique, (7) les conditions aux limites, (8) l’anisotropie, et (9) l’échelle. Les développements récents des réseaux de fractures en 3-D sont brièvement revus afin de souligner l’importance de l’utilisation de modèles 3-D dans les recherches futures.

Resumen

Las redes de fracturas juegan un papel significativo en la conducción del flujo del fluido y en el transporte de solutos en las masas de roca fracturada, en comparación con el de la matriz de la roca. La estimación precisa de la permeabilidad de las redes de fracturas ayudaría a los investigadores e ingenieros a evaluar mejor el rendimiento de los proyectos relacionados con el flujo de fluido en masas de rocas fracturadas. Este estudio proporciona una revisión de trabajos previos que se han centrado en la estimación de la permeabilidad equivalente de redes bidimensionales (2-D) discretas de fractura (dfns) teniendo en cuenta las influencias de las propiedades geométricas de las masas de rocas fracturadas. Se resumen las expresiones matemáticas para los efectos de los nueve parámetros importantes que impactan significativamente sobre la permeabilidad equivalente de dfns, incluyendo (1) la distribución de la longitud de la fractura, (2) la distribución de las aberturas, (3) la rugosidad de la superficie de fractura, (4) las fracturas cerradas, (5) el número de intersecciones, (6) el gradiente hidráulico, (7) la tensión límite, (8) la anisotropía, y (9) la escala. Se revisa brevemente el reciente desarrollo de redes de fracturas 3-D para resaltar la importancia de la utilización de modelos 3-D en futuras investigaciones.

摘要

裂隙网络相对于岩石基质对裂隙岩体内流体流动和污染物运移等性质有着更为重要的影响。准确预测裂隙网络的渗透系数将有助于研究人员更好的评估与裂隙岩体渗流相关的各项工程的安全稳定性。本综述回顾了前人利用裂隙岩体几何信息计算二维离散裂隙网络渗透系数的相关研究,讨论了裂隙网络模型中9个重要参数对渗透系数数学表达式的影响,这9个参数包括:(1) 裂隙长度分布, (2) 裂隙开度分布, (3) 裂隙表面粗糙度, (4) 裂隙断头, (5) 交点数量, (6) 水力梯度, (7) 边界应力, (8) 各向异性, (9) 模型尺寸。本文也介绍了三维裂隙网络模型的发展现状,并强调了在将来的工作中采用三维模型来评估渗透系数的重要性。

Resumo

Redes fraturadas desempenham um papel mais significante no fluxo de condução do fluido e no transporte de soluto na massa da rocha fraturada, comparado com o mesmo na rocha matriz. Estimativa precisa da permeabilidade das redes de fratura pode ajudar pesquisadores e engenheiros a melhor avaliar o desempenho dos processos associados com o fluxo de fluido na massa de rocha fraturada. Esse estudo fornece uma revisão de trabalhos anteriores que focaram na estimativa de permeabilidade de redes de fratura discretas (RFDs) bidimensionais (2-D) considerando a influência de propriedades geométricas de massa de rochas fraturadas. As expressões matemáticas para os efeitos de nove parâmetros importantes de impacto significante da permeabilidade equivalente das RFDs foram resumidas, incluindo (1) distribuição no comprimento da fratura, (2) distribuição da abertura, (3) rugosidade da superfície da fratura, (4) final da fratura, (5) número de intersecções, (6) gradiente hidráulico, (7) stress do limite, (8) anisotropia e (9) escala. Desenvolvimentos recentes de redes de fraturas 3-D foram brevemente revisados para destacar a importância da utilização de modelos 3-D em pesquisas futuras.

Appendix

Nomenclature

A :

Cross-sectional area

A′:

First coefficient in Forchheimer’s law-based function

B :

Dimensionless constant of proportionality

B(k):

Average spacing of the kth set of fractures

B′:

Second coefficient in Forchheimer’s law-based function

C :

Dimensional function

C _r :

Contact ratio

D _T :

Fractal dimension of nonlinear streamline of fluid flow

D _TP :

Fractal dimension for tortuosity of tortuous capillaries

D _f :

Fractal dimension of fracture backbone

D _p :

Fractal dimension for the size distribution of capillaries/pores

E :

Mechanical aperture

E _M :

Mean mechanical aperture

J :

Hydraulic gradient

J _c :

Critical hydraulic gradient

K :

Permeability of a fracture network

K ₀ :

Initial permeability of a fracture network

K _m :

Matrix permeability

K _t :

Permeability of a dual-porosity medium

L :

Size of a fracture system

M :

Number of sampling points

M′:

Total number of fracture sets

N :

Total number of fractures

N _i :

Number of intersections

P _ij :

Fracture geometry tensor

Q :

Macroscopic flow rate

R :

Random number

T :

Travel time of particle through a fracture network

T _f :

Fracture transmissivity

W :

A parameter for charactering connectivity of a fracture network

X ₁ :

Connectivity index

X ₂ :

Box-counting dimension of fracture lines

X ₃ :

Hydraulic conductivity

Z ₂ :

Roughness coefficient

a ₁ :

Power law exponent for fracture number-length relationship

a ₂ :

Power law exponent for shear displacement-length relationship

b :

Second moment of the lognormal distribution

b′:

Half aperture

d _c :

Fracture center density

e :

Hydraulic aperture

e _a :

Lower aperture limit

e _b :

Upper aperture limit

e _max :

Maximum opening displacement

e ^(m) :

The mth fracture aperture

g′:

Error function

g :

Gravitational acceleration

k _f :

Equivalent hydraulic conductivity of the fth set of fractures

k _n :

Normal stiffness

l :

Fracture length

l ₀ :

Straight length of a capillary

l _max :

Maximum fracture length

l _min :

Minimum fracture length

l _t :

Tortuous length of a fracture

n :

Number of fractures

n _f :

Unit vector normal to the fth set of fractures

p :

Probability coefficient

p ^* :

Generalized expression of a percolation

p _c :

Percolation threshold

q :

Flow rate of a single fracture

s _f :

Change in spacing

u _s :

Shear displacement

u _sp :

Peak shear displacement

l _M :

Arithmetic mean of the length of all fractures

$\left[\overline{K}\right]$ :

Equivalent permeability tensor

Greek letters

v ₁ :

First coefficient depending on the Euclidean dimension of the system

v ₂ :

A parameter for characterizing fracture length

ΔP :

Hydraulic pressure difference

Δε _zf :

Increment of the normal strain

Ω :

Fracture coordinate transformation coefficient

α ₁ :

Proportionality coefficient for fracture number-length relationship

α ₂ :

Proportionality coefficient in the fractal model

α ₃ :

Proportionality coefficient for shear displacement-length relationship

α ₄ :

Proportionality coefficient for opening displacement-length relationship

α′:

Fracture azimuth

δ ₁ :

Relative flow rate deviation rate

δ ₂ :

Relative time deviation rate

δ _ij :

Kroneker delta

δ _max :

Maximum shear displacement

ζ :

Overall connectivity of a fracture network

θ :

Fracture dip angle

λ :

Capillary/pore diameter

λ _max :

Maximum capillary/pore diameter

λ′:

Coefficient of aperture

λ′′:

A dimensionless scalar

μ :

Fluid viscosity

ξ :

Characteristic scale

ρ :

Fluid density

ρ _c :

Characteristic exponent

ρ′:

Dimensionless fracture density

σ _apert :

Standard deviation of mean mechanical aperture

σ _n :

Normal stress

σ _nc :

Critical normal stress

σ _slope :

Standard deviation of local slope of fracture surface

φ :

Fracture porosity

φ _i :

Equivalent porosity in i-direction of the Cartesian coordinates

ψ :

Second coefficient depending on the Euclidean dimension of the system

ω :

Upper and lower bounds between −1 and 1

Abbreviations

BEM:

Boundary element method

CEM:

Composite element method

DEM:

Distinct element method

DFN:

Discrete fracture network

EFNP:

Equivalent fracture network permeability

FEMDEM:

Finite-discrete element method

FVM:

Finite volume method

HM:

Hydro mechanical

JRC:

Joint roughness coefficient

LEFM:

Linear elastic fracture mechanics

MHFE:

Mixed hybrid finite element

NS:

Navier–Stokes

REV:

Representative elementary volume

Re:

Reynolds number

THM:

Thermo-hydro-mechanical

THMC:

Thermo-hydro-mechanical-chemical