Cubic law with aperture-length correlation: implications for network scale fluid flow

Abstract

Previous studies have computed and modeled fluid flow through fractured rock with the parallel plate approach where the volumetric flow per unit width normal to the direction of flow is proportional to the cubed aperture between the plates, referred to as the traditional cubic law. When combined with the square root relationship of displacement to length scaling of opening-mode fractures, total flow rates through natural opening-mode fractures are found to be proportional to apertures to the fifth power. This new relationship was explored by examining a suite of flow simulations through fracture networks using the discrete fracture network model (DFN). Flow was modeled through fracture networks with the same spatial distribution of fractures for both correlated and uncorrelated fracture length-to-aperture relationships. Results indicate that flow rates are significantly higher for correlated DFNs. Furthermore, the length-to-aperture relations lead to power-law distributions of network hydraulic conductivity which greatly influence equivalent permeability tensor values. These results confirm the importance of the correlated square root relationship of displacement to length scaling for total flow through natural opening-mode fractures and, hence, emphasize the role of these correlations for flow modeling.

Résumé

Des études antérieures ont modélisé un flux normal à la fracturation d’une roche assimilée à des plaques parallèles et calculé qu’il est proportionnel au cube de l’ouverture des plaques, par analogie avec la classique loi cubique. Si l’on introduit la racine carrée du déplacement rapporté à l’ouverture des fractures, on montre que le flux total à travers les fractures est proportionnel à la puissance cinq de l’ouverture. On a établi cette nouvelle relation en examinant une série de simulations de flux à travers des réseaux de fractures avec le modèle réseau de fractures discrètes (DFN). Le flux a été modélisé sur des réseaux présentant la même distribution de fractures, d’une part pour des longueurs de fractures corrélées avec les ouvertures, d’autre part non corrélées. Les résultats indiquent que les flux sont sensiblement plus élévés pour les réseaux DFN corrélés. De plus, les relations entre longueur et ouverture conduisent à des distributions exponentielles de la conductivité hydraulique du réseau, qui influencent grandement le tenseur de perméabilité équivalente. Ces résultats confirment l’importance de la racine carrée du déplacement rapportée au rapport d’ouverture des fractures pour le calcul du flux total et par suite met en évidence le rôle de ces corrélations dans la modélisation d’un flux.

Resumen

Estudios previos han calculado y modelado el flujo de fluidos a través de rocas fracturadas con una aproximación a placas paralelas donde el flujo volumétrico por unidad de ancho normal a la dirección de flujo es proporcional al cubo de la abertura entre las placas, tradicionalmente denominada ley cúbica. Al combinar la raíz cuadrada con la escala de la relación de desplazamiento a la longitud de abertura de fracturas de modo abierto natural, los ritmos totales de flujo a través de las fracturas resultan proporcionales a la quinta potencia de las aberturas. Esta nueva relación se exploró examinando una serie de simulaciones de flujo a través de redes de fracturas usando el modelo de redes de fracturas discretas (DFN). El flujo fue modelado a través de redes de fracturas con la misma distribución espacial de fracturas, tanto para las relaciones entre las longitudes como para las aberturas correlacionadas y no correlacionadas. Los resultados indican que los flujos son significativamente mayores para DFNs correlacionados. Además, la relación longitud - abertura conduce a distribuciones de potencias, de conductividad hidráulica de red, las cuales influyen grandemente en los valores equivalente del tensor de permeabilidad. Estos resultados confirman la importancia de la escala en la relación de la raíz cuadrada correlacionada del desplazamiento a la longitud para el flujo total a través de fracturas naturales de modo abierto y, por lo tanto, enfatiza el rol de estas correlaciones para el modelado de flujo.

Resumo

Estudos anteriores calcularam e modelaram o escoamento de um fluido em rochas fracturadas com a aproximação das placas paralelas, onde o fluxo volumétrico por unidade de largura normal à direcção do escoamento é proporcional ao cubo da abertura entre as placas, referido como a tradicional lei cúbica. Quando combinada com a relação da raiz quadrada do deslocamento com o comprimento da abertura das fracturas, observa-se que o fluxo total através das aberturas é proporcional às aberturas elevadas à quinta potência. Esta nova relação foi explorada através da análise de um conjunto de simulações do escoamento através de uma rede de fracturas utilizando um modelo de rede discreta de fracturas (discrete fracture network model - DFN). O escoamento foi modelado através de uma rede de fracturas com a mesma distribuição espacial para ambas as relações de comprimento-abertura correlacionadas e não correlacionadas. Os resultados indicam que os escoamentos são significativamente superiores para DFNs correlacionados. Além disso, as relações comprimento-abertura conduziram a leis de distribuição exponencial de redes de condutividade hidráulica que influenciam assinalavelmente os valores dos tensores de permeabilidade equivalente. Estes resultados confirmam a importância da correlação da relação da raiz quadrada do deslocamento com a escala de comprimento para um escoamento total através de fracturas e, assim, enfatiza o papel dessas correlações na modelação do escoamento.

Appendices

Appendix 1: Notation

a :

Fracture length distribution exponent

α :

Proportionality coefficient (opening-mode fractures), m^1/2

b :

Aperture, m

D :

Displacement, m

D _avg :

Average opening displacement, m

D _max :

Maximum opening displacement, m

∆ σ _I :

Opening-mode driving stress, Pa

E :

Young’s modulus, Pa

θ :

Fracture orientation

g :

Acceleration of gravity, m/s²

γ :

Proportionality coefficient (faults)

H :

Height of parallel plates, fracture height, m

∇h :

Hydraulic head gradient

∇H _x :

Head gradient in x-direction

∇H _y :

Head gradient in y-direction

I₀(ĸ):

Modified Bessel function of order zero

K _Ic :

Fracture toughness, MPa m^1/2

K :

Hydraulic conductivity, m/s

$\overline{\overline k}$ :

Permeability tensor, m²

K _xx :

Principal direction of permeability tensor, m²

K _xy :

Principal direction of permeability tensor, m²

K _yx :

Cross direction of permeability tensor, m²

K _yy :

Cross direction of permeability tensor, m²

κ :

Variation in fracture orientation

L :

Fracture length, m

L _min :

Minimum fracture length, m

μ :

Viscosity, Pa s

Q :

Volumetric discharge, m³/s

q _xx :

Principal direction of specific discharge, m/s

q _yx :

Cross direction of specific discharge, m/s

q _yx :

Cross direction of specific discharge, m/s

q _yy :

Principal direction of specific discharge, m/s

ρ :

Fluid density, kg/m³

U :

Random variable for fracture length computation

V _f :

Fracture volume, m³

ν :

Poisson’s ratio

W :

Length of parallel plates

ω :

Mean fracture orientation

Appendix 2

The fault datasets presented in Fig. 3 are taken from Schultz et al. (2008a) and references therein. The opening-mode fracture datasets in Fig. 4 consist of seven previously reported datasets from Olson (2003) and Schultz et al. (2008a, b). Three newly acquired datasets were measured in outcrops of the Sierra Nevada batholith around Lake Tahoe, CA.

A set of NW/SE striking vertical dikes was carefully measured in terms of their length and opening displacement in exposures of glacially polished Mesozoic granitoids of the Donner Summit Pluton, near Donner Pass, CA. Two types of dikes were observed. Simple dikes comprise the majority of dikes in this region and are of granodioritic composition. The second type of dikes, referred to as complex dikes consist of two phases, an outer pegmatitic phase and an inner phase of a more mafic composition (Ward 1993). This dataset consists of a total of 28 D/L measurements ranging from 1.13 to 50.5 m in length.

Glacially polished granitoids are also present west of Emerald Bay State Park, CA. Here, a set of E/W striking dikes of pegmatitic composition and another set of steeply NE\SW striking quartz-filled veins was measured between Cascade Lake and Granite Lake. Both sets each include 14 individual measurements of fracture length and opening displacement. Measured dike lengths range between 1.38 and 48 m. Measured veins have lengths between 2.06 and 11.42 m. Measurements were taken with a 30-m steel tape, displacements and lengths have measurement uncertainties of ±0.5 and ±10 mm, respectively.